(q,t)-hook formula for Tailed Insets and a Macdonald polynomial identity

. . . . .

...

(

q

,

t

)

-hook formula for Tailed Insets and a

Macdonald polynomial identity

Masao Ishikawa† †_{Okayama University}

Algebraic and Enumerative Combinatorics in Okayama February 23, 2018

Okayama University

. . . .

.. Abstract

Okada presented a conjecture on(q,t)-hook formula for general d-complete posets in the paper, Soichi Okada,(q,t)-Deformations of multivariate hook product formulae, J. Algebr. Comb. (2010) 32, 399 – 416. We consider the Tailed Inset case, and reduce the conjectured identity to an indentity of the Macodonald polynomials rephrasing Okada’s (q,t)-weights via Pieri coefficients of the Macodonald polynomials. Joint work with Frederic Jouhet (University of Lyon I).

. . . .

.. References

In this talk

...

1 _{M. Ishikawa, “(q},_{t)-hook formula for Birds and Banners”,}

arXiv:1302.1968 [math.CO].

...

2 _{S. Okada, “(q},_{t)-deformations of multivariate hook product} formulae”,arXiv:0909.0086 [math.CO]J. Algebraic Combin. 32 (2010), 399-416.

...

3 _{R. Proctor, “Dynkin diagram classication of}λ-minusule Bruhat lattices and of d-complete posets”, J. Algebraic Combin. 9 (1999), 61 – 94.

...

4 _{M. Vuleti´c, “A generalization of Macmahon’s formula”,}

arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer. Math. Soc. 361 (2009), 2789-2804.

...

5 _{S.O. Warnaar, “Rogers-Szeg ¨o polynomials and}

Hall-Littlewood symmetric functions”, J. Algebra 303 (2006), 810–830.

. . . .

.. References

In this talk ...

1 _{M. Ishikawa, “(q},_{t)-hook formula for Birds and Banners”,} arXiv:1302.1968 [math.CO].

...

2 _{S. Okada, “(q},_{t)-deformations of multivariate hook product}

formulae”,arXiv:0909.0086 [math.CO]J. Algebraic

Combin. 32 (2010), 399-416.

...

3 _{R. Proctor, “Dynkin diagram classication of}λ-minusule Bruhat lattices and of d-complete posets”, J. Algebraic Combin. 9 (1999), 61 – 94.

...

4 _{M. Vuleti´c, “A generalization of Macmahon’s formula”,} arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer. Math. Soc. 361 (2009), 2789-2804.

...

5 _{S.O. Warnaar, “Rogers-Szeg ¨o polynomials and}

Hall-Littlewood symmetric functions”, J. Algebra 303 (2006), 810–830.

. . . .

.. References

In this talk ...

1 _{M. Ishikawa, “(q},_{t)-hook formula for Birds and Banners”,} arXiv:1302.1968 [math.CO].

...

2 _{S. Okada, “(q},_{t)-deformations of multivariate hook product} formulae”,arXiv:0909.0086 [math.CO]J. Algebraic Combin. 32 (2010), 399-416.

...

3 _{R. Proctor, “Dynkin diagram classication of}λ_{-minusule Bruhat}

lattices and of d-complete posets”, J. Algebraic Combin. 9 (1999), 61 – 94.

...

4 _{M. Vuleti´c, “A generalization of Macmahon’s formula”,} arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer. Math. Soc. 361 (2009), 2789-2804.

...

5 _{S.O. Warnaar, “Rogers-Szeg ¨o polynomials and}

Hall-Littlewood symmetric functions”, J. Algebra 303 (2006), 810–830.

. . . .

.. References

In this talk ...

1 _{M. Ishikawa, “(q},_{t)-hook formula for Birds and Banners”,} arXiv:1302.1968 [math.CO].

...

2 _{S. Okada, “(q},_{t)-deformations of multivariate hook product} formulae”,arXiv:0909.0086 [math.CO]J. Algebraic Combin. 32 (2010), 399-416.

...

3 _{R. Proctor, “Dynkin diagram classication of}λ_{-minusule Bruhat} lattices and of d-complete posets”, J. Algebraic Combin. 9 (1999), 61 – 94.

...

4 _{M. Vuleti´c, “A generalization of Macmahon’s formula”,}

arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer.

Math. Soc. 361 (2009), 2789-2804.

...

5 _{S.O. Warnaar, “Rogers-Szeg ¨o polynomials and}

Hall-Littlewood symmetric functions”, J. Algebra 303 (2006), 810–830.

. . . .

.. References

In this talk ...

1 _{M. Ishikawa, “(q},_{t)-hook formula for Birds and Banners”,} arXiv:1302.1968 [math.CO].

...

2 _{S. Okada, “(q},_{t)-deformations of multivariate hook product} formulae”,arXiv:0909.0086 [math.CO]J. Algebraic Combin. 32 (2010), 399-416.

...

3 _{R. Proctor, “Dynkin diagram classication of}λ_{-minusule Bruhat} lattices and of d-complete posets”, J. Algebraic Combin. 9 (1999), 61 – 94.

...

4 _{M. Vuleti´c, “A generalization of Macmahon’s formula”,} arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer. Math. Soc. 361 (2009), 2789-2804.

...

5 _{S.O. Warnaar, “Rogers-Szeg ¨o polynomials and}

Hall-Littlewood symmetric functions”, J. Algebra 303 (2006), 810–830.

. . . .

.. Additional References

Further references:

...

6 _{G. Gasper, “Rogers’ linearization formula for the continuous}

q-ultraspherical polynomials and quadratic transformation formulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071.

...

7 _{I. G. Macdonald, Symmetric Functions and Hall Polynomials} (2nd ed.), Oxford Univ. Press, (1995).

...

8 _{G. Gasper and M. Rahman, Basic Hypergeometric Series} (2nd ed.), Cambridge Univ. Press, (1990, 2004).

...

9 _{R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nd}

ed.), Cambridge Univ. Press, (2001, 2012). ...

10 _{R. Stanley, Ordered Structures and Partitions, Memoirs AMS}

no. 119 (1972).

. . . .

.. Additional References

Further references: ...

6 _{G. Gasper, “Rogers’ linearization formula for the continuous} q-ultraspherical polynomials and quadratic transformation formulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071.

...

7 _{I. G. Macdonald, Symmetric Functions and Hall Polynomials}

(2nd ed.), Oxford Univ. Press, (1995).

...

8 _{G. Gasper and M. Rahman, Basic Hypergeometric Series} (2nd ed.), Cambridge Univ. Press, (1990, 2004).

...

9 _{R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nd} ed.), Cambridge Univ. Press, (2001, 2012).

...

10 _{R. Stanley, Ordered Structures and Partitions, Memoirs AMS}

no. 119 (1972).

. . . .

.. Additional References

Further references: ...

6 _{G. Gasper, “Rogers’ linearization formula for the continuous} q-ultraspherical polynomials and quadratic transformation formulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071. ...

7 _{I. G. Macdonald, Symmetric Functions and Hall Polynomials} (2nd ed.), Oxford Univ. Press, (1995).

...

8 _{G. Gasper and M. Rahman, Basic Hypergeometric Series}

(2nd ed.), Cambridge Univ. Press, (1990, 2004).

...

9 _{R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nd} ed.), Cambridge Univ. Press, (2001, 2012).

...

10 _{R. Stanley, Ordered Structures and Partitions, Memoirs AMS} no. 119 (1972).

. . . .

.. Additional References

Further references: ...

6 _{G. Gasper, “Rogers’ linearization formula for the continuous} q-ultraspherical polynomials and quadratic transformation formulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071. ...

7 _{I. G. Macdonald, Symmetric Functions and Hall Polynomials} (2nd ed.), Oxford Univ. Press, (1995).

...

8 _{G. Gasper and M. Rahman, Basic Hypergeometric Series} (2nd ed.), Cambridge Univ. Press, (1990, 2004).

...

9 _{R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nd}

ed.), Cambridge Univ. Press, (2001, 2012).

...

10 _{R. Stanley, Ordered Structures and Partitions, Memoirs AMS} no. 119 (1972).

. . . .

.. Additional References

Further references: ...

6 _{G. Gasper, “Rogers’ linearization formula for the continuous} q-ultraspherical polynomials and quadratic transformation formulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071. ...

7 _{I. G. Macdonald, Symmetric Functions and Hall Polynomials} (2nd ed.), Oxford Univ. Press, (1995).

...

8 _{G. Gasper and M. Rahman, Basic Hypergeometric Series} (2nd ed.), Cambridge Univ. Press, (1990, 2004).

...

9 _{R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nd} ed.), Cambridge Univ. Press, (2001, 2012).

...

10 _{R. Stanley, Ordered Structures and Partitions, Memoirs AMS}

no. 119 (1972).

. . . .

Introduction

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72)

..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers.LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers.LetA (P)be the set of

P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. P-partitions

Apartially ordered set(also called aposet) is a set P with a binary relation “≤” which isantisymmetric,transitive, andreflexive.

.

Definition (Stanley ’72) ..

...

Let P be a poset. AP-partitionis a mapπ :P → Nsatisfying x≤y in P =⇒ π(x)≥ π(y)inN,

whereNis the set of nonnegative integers. LetA (P)be the set of P-partitions. . Example (P-partitions) .. ... 1 0 1 2

. . . .

.. (Shifted) diagrams

Apartitonis a nonincreasing sequenceλ = (λ1, λ2, . . . ) of nonnegative integers with finitely manyλiunequal to zero. Thelengthandweightof

λ, denoted byℓ(λ)and|λ|, are the number and sum of the non-zeroλi respectively. Astrict partitionis a partition in which its parts are strictly decreasing. Ifλ is a partition (resp. strict partition), then itsdiagram D(λ)(resp.shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1≤ j ≤ λi}

S(λ) = { (i, j) ∈ Z2 : i≤ j ≤ λi+ i− 1 }.

.

Example (The diagram and shifted diagram forλ = (4,3,1))

. .

. ...

D(λ) = S(λ) =

. . . .

.. (Shifted) diagrams

Apartitonis a nonincreasing sequenceλ = (λ1, λ2, . . . ) of nonnegative integers with finitely manyλiunequal to zero. Thelengthandweightof

λ, denoted byℓ(λ)and|λ|, are the number and sum of the non-zeroλi

respectively. Astrict partitionis a partition in which its parts are strictly decreasing. Ifλ is a partition (resp. strict partition), then itsdiagram D(λ)(resp.shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1≤ j ≤ λi} S(λ) = { (i, j) ∈ Z2 : i≤ j ≤ λi+ i− 1 }.

.

Example (The diagram and shifted diagram forλ = (4,3,1))

..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) diagrams

Apartitonis a nonincreasing sequenceλ = (λ1, λ2, . . . ) of nonnegative integers with finitely manyλiunequal to zero. Thelengthandweightof

λ, denoted byℓ(λ)and|λ|, are the number and sum of the non-zeroλi

respectively. Astrict partitionis a partition in which its parts are strictly decreasing. Ifλ is a partition (resp. strict partition), then itsdiagram D(λ)(resp.shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1≤ j ≤ λi} S(λ) = { (i, j) ∈ Z2 : i≤ j ≤ λi+ i− 1 }.

.

Example (The diagram and shifted diagram forλ = (4,3,1))

..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) diagrams

Apartitonis a nonincreasing sequenceλ = (λ1, λ2, . . . ) of nonnegative integers with finitely manyλiunequal to zero. Thelengthandweightof

λ, denoted byℓ(λ)and|λ|, are the number and sum of the non-zeroλi

respectively. Astrict partitionis a partition in which its parts are strictly decreasing. Ifλ is a partition (resp. strict partition), then itsdiagram D(λ)(resp.shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1≤ j ≤ λi} S(λ) = { (i, j) ∈ Z2 : i≤ j ≤ λi+ i− 1 }.

.

Example (The diagram and shifted diagram forλ = (4,3,1))

..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) diagrams

Apartitonis a nonincreasing sequenceλ = (λ1, λ2, . . . ) of nonnegative integers with finitely manyλiunequal to zero. Thelengthandweightof

λ, denoted byℓ(λ)and|λ|, are the number and sum of the non-zeroλi

respectively. Astrict partitionis a partition in which its parts are strictly decreasing. Ifλ is a partition (resp. strict partition), then itsdiagram D(λ)(resp.shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1≤ j ≤ λi} S(λ) = { (i, j) ∈ Z2 : i≤ j ≤ λi+ i− 1 }.

.

Example (The diagram and shifted diagram forλ = (4,3,1)) ..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) shapes

A diagram D(λ)or a shifted diagram S(λ)is regarded as a poset by defining its order structure by

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

By this order the poset represented by a diagram P=D(λ)is called ashape, and the posets P =S(λ)is calledshifted shapes. .

Example (The shape and shifted shape forλ = (4,3,1))

..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) shapes

A diagram D(λ)or a shifted diagram S(λ)is regarded as a poset by defining its order structure by

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

By this order the poset represented by a diagram P=D(λ)is called ashape, and the posets P =S(λ)is calledshifted shapes. .

Example (The shape and shifted shape forλ = (4,3,1)) ..

...

D(λ) = S(λ) =

.

Example (The shape and shifted shape forλ = (4,3,1))

..

...

D(λ) = S(λ) =

. . . .

.. (Shifted) shapes

A diagram D(λ)or a shifted diagram S(λ)is regarded as a poset by defining its order structure by

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

By this order the poset represented by a diagram P=D(λ)is called ashape, and the posets P =S(λ)is calledshifted shapes. .

Example (The shape and shifted shape forλ = (4,3,1)) ..

...

D(λ) = S(λ) =

. . . .

.. Hook

For a partition (resp. strict partition)λand a cell(i,j)∈D(λ)(resp. S(λ)), thehook at(i,j)in D(λ)(resp. S(λ)), is defined by

H_D(λ)(i,j)={(i,j)} ∪ {(i,l)∈D(λ) :l >j} ∪ {(k,j)∈D(λ) :k >i} (resp.

HS(λ)(i,j)={(i,j)} ∪ {(i,l)∈S(λ) :l >j}

∪ {(k,j)∈D(λ) :k >i} ∪ {(j+1,l)∈S(λ) :l >j}). .

Example (The hook at(1,2)in D(λ)and S(λ)forλ = (4,3,1))

..

...

D(λ) = S(λ) =

4 5

. . . .

.. Hook

For a partition (resp. strict partition)λand a cell(i,j)∈D(λ)(resp. S(λ)), thehook at(i,j)in D(λ)(resp. S(λ)), is defined by

H_D(λ)(i,j)={(i,j)} ∪ {(i,l)∈D(λ) :l >j} ∪ {(k,j)∈D(λ) :k >i} (resp.

HS(λ)(i,j)={(i,j)} ∪ {(i,l)∈S(λ) :l >j}

∪ {(k,j)∈D(λ) :k >i} ∪ {(j+1,l)∈S(λ) :l >j}). .

Example (The hook at(1,2)in D(λ)and S(λ)forλ = (4,3,1)) ..

...

D(λ) = S(λ) =

4 5

. . . .

.. Content and hook length

.

Definition ..

...

Thehook length at(i,j)is defined byhD(λ)(i,j)=|HD(λ)(i,j)|(resp. h_S(λ)(i,j)=|H_S(λ)(i,j)|). Furtherc(i,j)=j−i is called thecontent at(i,j).

.

Example (The hook lenghs in D(λ)and S(λ)forλ = (4,3,1))

.. ... D(λ) = S(λ) = 6 4 3 1 4 2 1 1 7 5 4 2 4 3 1 1 .

Example (The contents in D(λ)and S(λ)forλ = (4,3,1))

.. ... D(λ) = S(λ) = 0 1 2 3 −1 0 1 −2 0 1 2 3 0 1 2 0

. . . .

.. Content and hook length

.

Definition ..

...

Thehook length at(i,j)is defined byhD(λ)(i,j)=|HD(λ)(i,j)|(resp. h_S(λ)(i,j)=|H_S(λ)(i,j)|). Furtherc(i,j)=j−i is called thecontent at(i,j).

.

Example (The hook lenghs in D(λ)and S(λ)forλ = (4,3,1)) .. ... D(λ) = S(λ) = 6 4 3 1 4 2 1 1 7 5 4 2 4 3 1 1 .

Example (The contents in D(λ)and S(λ)forλ = (4,3,1))

.. ... D(λ) = S(λ) = 0 1 2 3 −1 0 1 −2 0 1 2 3 0 1 2 0

. . . .

.. Content and hook length

.

Definition ..

...

Thehook length at(i,j)is defined byhD(λ)(i,j)=|HD(λ)(i,j)|(resp. h_S(λ)(i,j)=|H_S(λ)(i,j)|). Furtherc(i,j)=j−i is called thecontent at(i,j).

.

Example (The hook lenghs in D(λ)and S(λ)forλ = (4,3,1)) .. ... D(λ) = S(λ) = 6 4 3 1 4 2 1 1 7 5 4 2 4 3 1 1 .

Example (The contents in D(λ)and S(λ)forλ = (4,3,1)) .. ... D(λ) = S(λ) = 0 1 2 3 −1 0 1 −2 0 1 2 3 0 1 2 0

. . . .

.. One Variable Hook Length Formula

.

Theorem (Frame-Robinson-Thrall ’54, Stanley ’72)) .. ... If P =D(λ)or S(λ), then we have ∑ π∈A (P) z|π| = ∏ (i,j)∈P 1 1−zhP(i,j),

where the sum on the left-hand side runs over all P-partitions, and |π| =∑x∈Pπ(x).

.

Example (An example of P-partition)

.. ... π = |π| =16 z|π|=z16 0 0 1 2 2 3 4 4

. . . .

.. One Variable Hook Length Formula

.

Theorem (Frame-Robinson-Thrall ’54, Stanley ’72)) .. ... If P =D(λ)or S(λ), then we have ∑ π∈A (P) z|π| = ∏ (i,j)∈P 1 1−zhP(i,j),

where the sum on the left-hand side runs over all P-partitions, and |π| =∑x∈Pπ(x).

.

Example (An example of P-partition) .. ... π = |π| =16 z|π|=z16 0 0 1 2 2 3 4 4

. . . .

.. Example of One Variable Hook Length Formula

.

Example (The shape forλ = (4,3,1)) .. ... D(λ) = D(λ) = π11π12π13π14 π21π22π23 π31 6 4 3 1 4 2 1 1 ∑ π∈A (D(λ)) z∑(i,j)∈D(λ)πi,j ₌ 1 (1−z)3₍₁−_z2₎₍₁−_z3₎₍₁−_z4₎2₍₁−_z6₎.

. . . .

.. Multivariate Hook Length Formula

.

Theorem (Gansner ’81, Sagan ’82) ..

...

Let. . . ,z−1,z0,z1,z2, . . . be variables. If P =D(λ)or S(λ), then

we have _∑ π∈A (P) zπ = ∏ (i,j)∈P 1 1−z[HP(i,j)] ,

where the sum on the left-hand side runs over all P-partitions, zπ =∏_(i,j)∈Pzc(i,j)πi,j andz[H]=∏(i,j)∈Hzc(i,j)for any finite subset H⊂ Z2. (Gansner used Hillman-Grassl ’76 algorithm.)

.

Example (An example of P-partition)

.. ... π = zπ =z₋₂4 z₋₁2 z₀3z₁4z2z₃2 0 0 1 2 2 3 4 4

. . . .

.. Multivariate Hook Length Formula

.

Theorem (Gansner ’81, Sagan ’82) ..

...

Let. . . ,z−1,z0,z1,z2, . . . be variables. If P =D(λ)or S(λ), then

we have _∑ π∈A (P) zπ = ∏ (i,j)∈P 1 1−z[HP(i,j)] ,

where the sum on the left-hand side runs over all P-partitions, zπ =∏_(i,j)∈Pzc(i,j)πi,j andz[H]=∏(i,j)∈Hzc(i,j)for any finite subset H⊂ Z2. (Gansner used Hillman-Grassl ’76 algorithm.)

.

Example (An example of P-partition) .. ... π = zπ =z₋₂4 z₋₁2 z₀3z₁4z2z₃2 0 0 1 2 2 3 4 4

. . . .

.. Example of Multivariate Hook Length Formula

.

Example (The shape forλ = (4,3,1)) .. ... D(λ) = D(λ) = π11π12π13π14 π21π22π23 π31 z0 z1 z2 z3 z₋₁ z0 z1 z₋₂ ∑ π∈A (P) z₋₂π31z₋₁π21z₀π11+π22z₁π12+π23zπ13₂ z₃π14 = 1 (1−z₋₂z₋₁z0z1z2z3)(1−z0z1z2z3)(1−z1z2z3)(1−z3) × 1 (1−z₋₂z₋₁z0z1)(1−z0z1)(1−z1)(1−z−2) .

. . . .

.. The Cauchy formula and the Littlewood formula

.

Therem (The Cauchy formula) ..

...

Let x= (x1, . . . ,xn)and y = (y1, . . . ,yn)are n-tuples of variables. Then we have ∑ λ s_λ(x)s_λ(y) = n ∏ i,j=1 1 1−xiyj. .

Therem (The Littlewood formula)

..

...

Let x= (x1, . . . ,xn)is an n-tuples of variables. Then we have

∑ λ s_λ(x) = n ∏ i=1 1 1−xi ∏ 1≤i<j≤n 1 1−xixj.

. . . .

.. The Cauchy formula and the Littlewood formula

.

Therem (The Cauchy formula) ..

...

Let x= (x1, . . . ,xn)and y = (y1, . . . ,yn)are n-tuples of variables. Then we have ∑ λ s_λ(x)s_λ(y) = n ∏ i,j=1 1 1−xiyj. .

Therem (The Littlewood formula) ..

...

Let x= (x1, . . . ,xn)is an n-tuples of variables. Then we have ∑ λ s_λ(x) = n ∏ i=1 1 1−xi ∏ 1≤i<j≤n 1 1−xixj.

. . . .

..

(

q

,

t

)

-hook formula

If P is a d-complete poset, then we have ∑

π∈A (P)

WP(π;q,t)zπ= ∏ (i,j)∈P

F(z[HP(i,j)];q,t),

where the sum on the left-hand side runs over all P-partitions, and

F(x;q,t)= (tx;q)∞ (x;q)_∞. .

Example (The shape forλ = (4,3,1))

.. ... D(λ) = D(λ) = π11π12π13π14 π21π22π23 π31 z0 z1 z2 z3 z₋₁ z0 z1 z₋₂ ∑ π∈A (P) WP(π;q,t)z₋₂π31z₋₁π21z₀π11+π22z₁π12+π23z₂π13z₃π14 =F(z₋₂z₋₁z0z1z2z3;q,t)F(z0z1z2z3;q,t)F(z1z2z3;q,t) ×F(z3;q,t)F(z−2z−1z0z1;q,t)F(z0z1;q,t)F(z1;q,t)F(z−2;q,t).

. . . .

..

(

q

,

t

)

-hook formula

.

Example (The shape forλ = (4,3,1)) .. ... D(λ) = D(λ) = π11π12π13π14 π21π22π23 π31 z0 z1 z2 z3 z₋₁ z0 z1 z₋₂ ∑ π∈A (P) WP(π;q,t)z₋₂π31z₋₁π21z₀π11+π22z₁π12+π23z₂π13z₃π14 =F(z₋₂z₋₁z0z1z2z3;q,t)F(z0z1z2z3;q,t)F(z1z2z3;q,t) ×F(z3;q,t)F(z−2z−1z0z1;q,t)F(z0z1;q,t)F(z1;q,t)F(z−2;q,t).

. . . .

.. Current situation

1 _{If P is (1) Shape or (2) Shfted Shape, the(q},_{t)-hook formula}

is proven in the paper by Okada(2010).

...

2 _{If P is (3) Bird or (6) Banner, the(q},_{t)-hook formula is proven}

by me (not yet published) 2013. We use Gasper’s identity. .

..

3 _{This talk is about the Tailed Inset case (not yet completed).}

. . . .

.. Current situation

1 _{If P is (1) Shape or (2) Shfted Shape, the(q},_{t)-hook formula}

is proven in the paper by Okada(2010).

...

2 _{If P is (3) Bird or (6) Banner, the(q},_{t)-hook formula is proven}

by me (not yet published) 2013. We use Gasper’s identity.

...

3 _{This talk is about the Tailed Inset case (not yet completed).}

. . . .

.. Current situation

1 _{If P is (1) Shape or (2) Shfted Shape, the}(q,_{t)-hook formula} is proven in the paper by Okada(2010).

...

2 _{If P is (3) Bird or (6) Banner, the(q},_{t)-hook formula is proven}

by me (not yet published) 2013. We use Gasper’s identity.

...

3 _{This talk is about the Tailed Inset case (not yet completed).}

. . . .

.. Current situation

1 _{If P is (1) Shape or (2) Shfted Shape, the}(q,_{t)-hook formula} is proven in the paper by Okada(2010).

...

2 _{If P is (3) Bird or (6) Banner, the(q},_{t)-hook formula is proven} by me (not yet published) 2013. We use Gasper’s identity.

...

3 _{This talk is about the Tailed Inset case (not yet completed).}

. . . .

.. The Cauchy type identity for Macdonald polynomials

.

Theorem ..

...

Let x= (x1, . . . ,xn)and y = (y1, . . . ,yn)are n-tuples of variables. Then we have ∑ λ P_λ(x;q,t)Q_λ(y;q,t) = n ∏ i,j=1 F(xiyj;q,t).

. . . .

.. Warnaar’s formula

where r(λ) is the number of rows of odd length.

. Further Corollary . . . ... ∑ λ w|λ|+r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ , ∑ λ w|λ|−r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (txi; q)_∞ (xi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ .

. . . .

.. Warnaar’s formula

where r(λ) is the number of rows of odd length. . Corollary .. ... ∑ λ wr(λ′)b_λel(q, t)P_λ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (txixj; q)∞ (xixj; q)_∞ .

Proof. Applying theF-algebra homomorphism wq,t to the above

identity. . Further Corollary .. ... ∑ λ w|λ|+r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ , ∑ λ w|λ|−r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (txi; q)_∞ (xi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ . Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Warnaar’s formula

where r(λ) is the number of rows of odd length. . Corollary .. ... ∑ λ wr(λ′)b_λel(q, t)P_λ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (txixj; q)∞ (xixj; q)_∞ .

Proof. Applying theF-algebra homomorphism wq,t to the above identity. . Further Corollary .. ... ∑ λ w|λ|+r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (twxixj; q)_∞ (wxixj; q)_∞, ∑ λ w|λ|−r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (txi; q)_∞ (xi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ . Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Warnaar’s formula

where r(λ) is the number of rows of odd length. . Further Corollary .. ... ∑ λ w|λ|+r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (twxi; q)_∞ (wxi; q)_∞ ∏ i<j (twxixj; q)∞ (wxixj; q)_∞ , ∑ λ w|λ|−r(λ′)2 bel λ(q, t)Pλ(x; q, t) = ∏ i≥1 (txi; q)_∞ (xi; q)_∞ ∏ i<j (twxixj; q)_∞ (wxixj; q)_∞.

. . . .

d-complete poset

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule}

heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of}

d-complete poset, which is a graded poset with d-complete

coloring. .

..

3 _{Proctor showed that any d-complete poset can be obtained}

from the 15irreducibleclasses byslant-sum.

. ..

4 _{Thed-complete coloringis important for the multivariate}

generating function. The content should be replaced by color for d-complete posets.

. ..

5 _Okada’s(q,_t)-weightWP(π;_q,_t) .

..

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule}

heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of}

d-complete poset, which is a graded poset with d-complete

coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained}

from the 15irreducibleclasses byslant-sum.

. ..

4 _{Thed-complete coloringis important for the multivariate}

generating function. The content should be replaced by color for d-complete posets.

. ..

5 _Okada’s(q,_t)-weightWP(π;_q,_t) .

..

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule} heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of}

d-complete poset, which is a graded poset with d-complete

coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained}

from the 15irreducibleclasses byslant-sum.

...

4 _{Thed-complete coloringis important for the multivariate}

generating function. The content should be replaced by color for d-complete posets.

. ..

5 _Okada’s(q,_t)-weightWP(π;_q,_t) .

..

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule} heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of} d-complete poset, which is a graded poset with d-complete coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained}

from the 15irreducibleclasses byslant-sum.

...

4 _{Thed-complete coloringis important for the multivariate}

generating function. The content should be replaced by color for d-complete posets.

...

5 _Okada’s(q,_t)-weightWP(π;_q,_t)

. ..

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule} heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of} d-complete poset, which is a graded poset with d-complete coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained} from the 15irreducibleclasses byslant-sum.

...

4 _{Thed-complete coloringis important for the multivariate}

generating function. The content should be replaced by color for d-complete posets.

...

5 _Okada’s(q,_t)-weightWP(π;_q,_t)

...

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule} heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of} d-complete poset, which is a graded poset with d-complete coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained} from the 15irreducibleclasses byslant-sum.

...

4 _{Thed-complete coloringis important for the multivariate} generating function. The content should be replaced by color for d-complete posets.

...

5 _Okada’s(q,_{t)-weight WP(}π;_q,_t)

...

6 _{Hook monomials for d-complete posets}

. . . .

.. d-complete poset

.

Contents of this section ..

...

...

1 _{The d-complete posets arise from the dominant minuscule} heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras.

...

2 _{Proctor gave completely combinatorial description of} d-complete poset, which is a graded poset with d-complete coloring.

...

3 _{Proctor showed that any d-complete poset can be obtained} from the 15irreducibleclasses byslant-sum.

...

4 _{Thed-complete coloringis important for the multivariate} generating function. The content should be replaced by color for d-complete posets.

...

5 _Okada’s(q,_t)-weightWP(π;_q,_t)

...

6 _{Hook monomials for d-complete posets}

. . . .

.. Double-tailed diamond poset

.

Definition ..

...

Thedouble-tailed diamond poset dk(1)is the poset depicted below: k−2 k−2 top side side bottom

Adk-intervalis an interval isomorphic to dk(1).

Ad_k−-interval(k ≥4) is an interval isomorphic to dk(1)− {top}.

Ad₃−-intervalconsists of three elements x, y and w such that

w is covered by x and y.

. . . .

.. Double-tailed diamond poset

.

Definition ..

...

The double-tailed diamond poset dk(1)is the poset depicted

below: k−2 k−2 top side side bottom

Adk-intervalis an interval isomorphic to dk(1).

Ad_k−-interval(k ≥4) is an interval isomorphic to dk(1)− {top}.

Ad₃−-intervalconsists of three elements x, y and w such that

w is covered by x and y.

. . . .

.. Double-tailed diamond poset

.

Definition ..

...

Thedouble-tailed diamond poset dk(1)is the poset depicted

below: k−2 k−2 top side side bottom

A dk-interval is an interval isomorphic to dk(1).

Ad_k−-interval(k ≥4) is an interval isomorphic to dk(1)− {top}.

Ad₃−-intervalconsists of three elements x, y and w such that

w is covered by x and y.

. . . .

.. Double-tailed diamond poset

.

Definition ..

...

Thedouble-tailed diamond poset dk(1)is the poset depicted

below: k−2 k−2 top side side bottom

Adk-intervalis an interval isomorphic to dk(1).

A d_k−-interval (k ≥4) is an interval isomorphic to dk(1)− {top}.

Ad₃−-intervalconsists of three elements x, y and w such that

w is covered by x and y.

. . . .

.. Double-tailed diamond poset

.

Definition ..

...

Thedouble-tailed diamond poset dk(1)is the poset depicted

below: k−2 k−2 top side side bottom

Adk-intervalis an interval isomorphic to dk(1).

Ad_k−-interval(k ≥4) is an interval isomorphic to dk(1)− {top}.

A d₃−-interval consists of three elements x, y and w such that

w is covered by x and y.

. . . .

.. Definition of d-complete poset

.

Definition ..

...

A poset P isd-completeif it satisfies the following three conditions for every k ≥3:

...

1 _{If I is a d}−

k-interval, then there exists an element v such that v

covers the maximal elements of I and I∪ {v}is a dk-interval. ...

2 _{If I = [w},_{v]is a dk-interval and the top v covers u in P, then}

u∈I.

...

3 _{There are no d}−

k-intervals which differ only in the minimal elements.

. . . .

.. Definition of d-complete poset

.

Definition ..

...

A poset P isd-completeif it satisfies the following three conditions for every k ≥3:

...

1 _{If I is a d}−

k-interval, then there exists an element v such that v

covers the maximal elements of I and I∪ {v}is a dk-interval.

...

2 _{If I = [w},_{v]is a dk-interval and the top v covers u in P, then}

u∈I.

...

3 _{There are no d}−

k-intervals which differ only in the minimal

elements.

. . . .

.. Definition of d-complete poset

.

Definition ..

...

A poset P isd-completeif it satisfies the following three conditions for every k ≥3:

...

1 _{If I is a d}−

k-interval, then there exists an element v such that v covers the maximal elements of I and I∪ {v}is a dk-interval.

...

2 _{If I = [w},_{v]is a dk-interval and the top v covers u in P, then}

u∈I.

...

3 _{There are no d}−

k-intervals which differ only in the minimal

elements.

. . . .

.. Definition of d-complete poset

.

Definition ..

...

A poset P isd-completeif it satisfies the following three conditions for every k ≥3:

...

1 _{If I is a d}−

k-interval, then there exists an element v such that v covers the maximal elements of I and I∪ {v}is a dk-interval. ...

2 _{If I = [w},_{v]is a dk-interval and the top v covers u in P, then} u∈I.

...

3 _{There are no d}−

k-intervals which differ only in the minimal

elements.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch

that r(x) =r(y) +1 if x covers y.

.

Fact

..

...

(a) Any connected d-complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15 families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch

that r(x) =r(y) +1 if x covers y.

.

Fact

..

...

(a) Any connected d-complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible

d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15 families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch

that r(x) =r(y) +1 if x covers y.

.

Fact

..

...

(a) Any connected d-complete poset is uniquely decomposed into

a slant sum of one-element posets and slant-irreducible

d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15 families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch that r(x) =r(y) +1 if x covers y.

. Fact ..

...

(a) Any connected d-complete poset is uniquely decomposed into

a slant sum of one-element posets and slant-irreducible

d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15

families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch that r(x) =r(y) +1 if x covers y.

. Fact ..

...

(a) Any connected d-complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible

d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15

families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Properties of d-complete posets

. Fact ..

...

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P isgraded, i.e., there exists a rank function r :P → Nsuch that r(x) =r(y) +1 if x covers y.

. Fact ..

...

(a) Any connected d-complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible d-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15 families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets, (5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9) Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12) Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . .

.. Examples

. . . .

.. 15 irreducible d-complete posets

. . . .

.. 15 irreducible d-complete posets

. . . .

.. 15 irreducible d-complete posets

. . . .

.. 15 irreducible d-complete posets

. . . .

.. Top Tree and d-Complete Coloring

.

Definition ..

...

For a connected d-complete poset P, we define itstop treeby putting

T ={x∈P : every y ≥x is covered by at most one other element} .

Fact

..

...

Let I be a set of colors such that#I = #T . Then a bijection c:T →I can be uniquely extended to a map c:P →I satisfying the following three conditions:

If x and y are incomparable, then c(x),c(y).

If an interval[w,v]is a chain, then the colors c(x)(x∈ [w,v]) are distinct.

If[w,v]is a dk-interval then c(w) =c(v).

Such a map c:P →I is called ad-complete coloring. Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Top Tree and d-Complete Coloring

.

Definition ..

...

For a connected d-complete poset P, we define itstop treeby putting

T ={x∈P : every y ≥x is covered by at most one other element} .

Fact ..

...

Let I be a set of colors such that#I = #T . Then a bijection c:T →I can be uniquely extended to a map c:P →I satisfying the following three conditions:

If x and y are incomparable, then c(x),c(y).

If an interval[w,v]is a chain, then the colors c(x)(x∈ [w,v])

are distinct.

If[w,v]is a dk-interval then c(w) =c(v).

Such a map c:P →I is called ad-complete coloring. Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Top Tree and d-Complete Coloring

.

Definition ..

...

For a connected d-complete poset P, we define itstop treeby putting

T ={x∈P : every y ≥x is covered by at most one other element} .

Fact ..

...

Let I be a set of colors such that#I = #T . Then a bijection c:T →I can be uniquely extended to a map c:P →I satisfying the following three conditions:

If x and y are incomparable, then c(x),c(y).

If an interval[w,v]is a chain, then the colors c(x)(x∈ [w,v])

are distinct.

If[w,v]is a dk-interval then c(w) =c(v).

Such a map c:P →I is called ad-complete coloring. Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Top Tree and d-Complete Coloring

.

Definition ..

...

For a connected d-complete poset P, we define itstop treeby putting

T ={x∈P : every y ≥x is covered by at most one other element} .

Fact ..

...

Let I be a set of colors such that#I = #T . Then a bijection c:T →I can be uniquely extended to a map c:P →I satisfying the following three conditions:

If x and y are incomparable, then c(x),c(y).

If an interval[w,v]is a chain, then the colors c(x)(x∈ [w,v])

are distinct.

If[w,v]is a dk-interval then c(w) =c(v).

Such a map c:P →I is called ad-complete coloring. Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Top Tree and d-Complete Coloring

.

Definition ..

...

For a connected d-complete poset P, we define itstop treeby putting

T ={x∈P : every y ≥x is covered by at most one other element} .

Fact ..

...

Let I be a set of colors such that#I = #T . Then a bijection c:T →I can be uniquely extended to a map c:P →I satisfying the following three conditions:

If x and y are incomparable, then c(x),c(y).

If an interval[w,v]is a chain, then the colors c(x)(x∈ [w,v]) are distinct.

If[w,v]is a dk-interval then c(w) =c(v).

Such a map c:P →I is called ad-complete coloring. Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . .

.. Top Tree and d-Complete Coloring

.

Example ..

...

Top Tree and d-Complete Coloringof d5-interval

. . . .

.. Top Tree and d-Complete Coloring

.

Example ..

...

Top Treeand d-Complete Coloringof d5-interval

. . . .

.. Top Tree and d-Complete Coloring

.

Example ..

...

Top Tree and d-Complete Coloring of d5-interval

. . . .

.. Top Tree and d-Complete Coloring

.

Example ..

...

Top Tree and d-Complete Coloring of d5-interval

. . . .

.. Shapes

Let r be a postive integer, andα = (α1, . . . , αr)and β = (β1, . . . , βr)be strict partitions such that

α1 > · · · > αr ≥0, β1> · · · > βr ≥0, Let P be the set P=PL∪PR of lattice points inZ2, where

PR ={ (i,j) : 1≤i ≤j ≤ αi+i−1(1≤i ≤r)},

PL ={ (i,j) : 1≤j ≤i≤ βj+j−1(1≤j eqr)},

We regard P as a poset by defining the order relation (i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2. We call this poset ashapeand denote it by P =P1(α, β).

. . . .

.. Shapes

Let r be a postive integer, andα = (α1, . . . , αr)and β = (β1, . . . , βr)be strict partitions such that

α1 > · · · > αr ≥0, β1> · · · > βr ≥0, Let P be the set P=PL∪PR of lattice points inZ2, where

PR ={ (i,j) : 1≤i ≤j ≤ αi+i−1(1≤i ≤r)}, PL ={ (i,j) : 1≤j ≤i≤ βj+j−1(1≤j eqr)}, We regard P as a poset by defining the order relation

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2. We call this poset ashapeand denote it by P =P1(α, β).

. . . .

.. Shapes

Let r be a postive integer, andα = (α1, . . . , αr)and β = (β1, . . . , βr)be strict partitions such that

α1 > · · · > αr ≥0, β1> · · · > βr ≥0, Let P be the set P=PL∪PR of lattice points inZ2, where

PR ={ (i,j) : 1≤i ≤j ≤ αi+i−1(1≤i ≤r)}, PL ={ (i,j) : 1≤j ≤i≤ βj+j−1(1≤j eqr)}, We regard P as a poset by defining the order relation

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2. We call this poset ashapeand denote it by P =P1(α, β).

. . . .

.. Shapes

(1,1)

. . . .

.. Shifted Shapes

Let r be a postive integer, andα = (α1, . . . , αr)be a strict partition such that

α1 > · · · > αr ≥0. Define theshifted shapeP =P2(α)by

P ={ (i,j) : i ≤j ≤ αi+i−1(1≤i ≤r)}.

We regard it as a poset by defining its order structure (i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

. . . .

.. Shifted Shapes

Let r be a postive integer, andα = (α1, . . . , αr)be a strict partition such that

α1 > · · · > αr ≥0. Define theshifted shapeP =P2(α)by

P ={ (i,j) : i ≤j ≤ αi+i−1(1≤i ≤r)}. We regard it as a poset by defining its order structure

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

. . . .

.. Shifted Shapes

Let r be a postive integer, andα = (α1, . . . , αr)be a strict partition such that

α1 > · · · > αr ≥0. Define theshifted shapeP =P2(α)by

P ={ (i,j) : i ≤j ≤ αi+i−1(1≤i ≤r)}. We regard it as a poset by defining its order structure

(i1,j1)≥(i2,j2)⇐⇒ i1≤i2and j1 ≤j2.

. . . .

.. Shifted Shape

(1,1)