The generalised Delta square conjecture

∗

The generalised Delta square conjecture

Anna Vanden Wyngaerd

joint work with Michele D’Adderio and Alessandro Iraci April 15, 2019

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,... I Kostka-Macdonald coefficients

H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,... I Kostka-Macdonald coefficients

H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,... I Kostka-Macdonald coefficients

H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,... I Kostka-Macdonald coefficients

H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,...

I Kostka-Macdonald coefficients H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,...

I Kostka-Macdonald coefficients H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

MacDonald Polynomials

Λ_C(q,t):=C(q, t)[X1, ..., XN]^SN =L∞ i=1Λⁿ

C(q,t)

I Whenn≥N, basis ofΛⁿ

C(q,t)include elementarye_λ, homogeneoush_λ, powerp_λ and Schurs_λsymmetric functions.

I {H˜λ|λ`n}(modified, Garsia & Haiman) Macdonald Polynomials: basis ofΛ_C(q,t)

I Applications in wide variety of subjects

I "Generalisation" of Hall-Littlewood, Jack polynomials,...

I Kostka-Macdonald coefficients H˜_µ=X

λ`n

K˜_λµ(q, t)s_λ

Macdonald Positivity Conjecture

K˜_λµ(q, t)∈N[q, t], i.e. the Macdonald polynomials areSchur positive

n! conjecture

Strategy to prove Schur positivity of Macdonald Polynomials

I Construction, for eachµ, a bi-graded moduleM_µ(Garsia Haiman module), affording regular representation ofS_n I H˜µis image of the bi-graded character of this module by

Frobenius characteristic map

I Garsia and Haiman reduced this to the problem of showing that Dim(M_µ) =n!

I Proved by Haiman in 2001, using tools from Algebraic Geometry

n! conjecture

Strategy to prove Schur positivity of Macdonald Polynomials I Construction, for eachµ, a bi-graded moduleM_µ(Garsia

Haiman module), affording regular representation ofS_n

I H˜µis image of the bi-graded character of this module by Frobenius characteristic map

I Garsia and Haiman reduced this to the problem of showing that Dim(M_µ) =n!

I Proved by Haiman in 2001, using tools from Algebraic Geometry

n! conjecture

Strategy to prove Schur positivity of Macdonald Polynomials I Construction, for eachµ, a bi-graded moduleM_µ(Garsia

Haiman module), affording regular representation ofS_n I H˜µis image of the bi-graded character of this module by

Frobenius characteristic map

I Garsia and Haiman reduced this to the problem of showing that Dim(M_µ) =n!

I Proved by Haiman in 2001, using tools from Algebraic Geometry

n! conjecture

Strategy to prove Schur positivity of Macdonald Polynomials I Construction, for eachµ, a bi-graded moduleM_µ(Garsia

Haiman module), affording regular representation ofS_n I H˜µis image of the bi-graded character of this module by

Frobenius characteristic map

I Garsia and Haiman reduced this to the problem of showing that Dim(M_µ) =n!

I Proved by Haiman in 2001, using tools from Algebraic Geometry

n! conjecture

Strategy to prove Schur positivity of Macdonald Polynomials I Construction, for eachµ, a bi-graded moduleM_µ(Garsia

Haiman module), affording regular representation ofS_n I H˜µis image of the bi-graded character of this module by

Frobenius characteristic map

I Garsia and Haiman reduced this to the problem of showing that Dim(M_µ) =n!

I Proved by Haiman in 2001, using tools from Algebraic Geometry

The Delta operators

Working on the Macdonald positivity conjecture, Garsia and Haiman introduced theS_N-moduleDH_nofdiagonal harmonics.

It turns out that

F(DH_n;q, t) =∇e_n

I ∇is the operator defined by∇He_µ:=T_µHe_µwhereT_µ∈N[q, t]. I TheDelta operators, for somef ∈Λare defined by

∆_fHe_µ:=f[B_µ(q, t)]He_µ and ∆⁰_fHe_µ:=f[B_µ(q, t)−1]He_µ, whereBµ∈N[q, t].

I OnΛ⁽ⁿ⁾,

∆en =∇ and ∆ek = ∆⁰_ek+ ∆⁰_ek−1

I Just a few weeks ago, Zabrocki found a module extending the diagonal harmonics, whose bi-graded Frobenius characteristic he conjectured to be∆⁰_en−k−1e_n.

The Delta operators

Working on the Macdonald positivity conjecture, Garsia and Haiman introduced theS_N-moduleDH_nofdiagonal harmonics.

It turns out that

F(DH_n;q, t) =∇e_n

I ∇is the operator defined by∇He_µ:=T_µHe_µwhereT_µ∈N[q, t].

I TheDelta operators, for somef ∈Λare defined by

∆_fHe_µ:=f[B_µ(q, t)]He_µ and ∆⁰_fHe_µ:=f[B_µ(q, t)−1]He_µ, whereBµ∈N[q, t].

I OnΛ⁽ⁿ⁾,

∆en =∇ and ∆ek = ∆⁰_ek+ ∆⁰_ek−1

I Just a few weeks ago, Zabrocki found a module extending the diagonal harmonics, whose bi-graded Frobenius characteristic he conjectured to be∆⁰_en−k−1e_n.

The Delta operators

Working on the Macdonald positivity conjecture, Garsia and Haiman introduced theS_N-moduleDH_nofdiagonal harmonics.

It turns out that

F(DH_n;q, t) =∇e_n

I ∇is the operator defined by∇He_µ:=T_µHe_µwhereT_µ∈N[q, t].

I TheDelta operators, for somef ∈Λare defined by

∆_fHe_µ:=f[B_µ(q, t)]He_µ and ∆⁰_fHe_µ:=f[B_µ(q, t)−1]He_µ, whereB_µ∈N[q, t].

I OnΛ⁽ⁿ⁾,

∆en =∇ and ∆ek = ∆⁰_ek+ ∆⁰_ek−1

I Just a few weeks ago, Zabrocki found a module extending the diagonal harmonics, whose bi-graded Frobenius characteristic he conjectured to be∆⁰_en−k−1e_n.

The Delta operators

Working on the Macdonald positivity conjecture, Garsia and Haiman introduced theS_N-moduleDH_nofdiagonal harmonics.

It turns out that

F(DH_n;q, t) =∇e_n

I ∇is the operator defined by∇He_µ:=T_µHe_µwhereT_µ∈N[q, t].

I TheDelta operators, for somef ∈Λare defined by

∆_fHe_µ:=f[B_µ(q, t)]He_µ and ∆⁰_fHe_µ:=f[B_µ(q, t)−1]He_µ, whereB_µ∈N[q, t].

I OnΛ⁽ⁿ⁾,

∆en =∇ and ∆ek = ∆⁰_ek+ ∆⁰_ek−1

I Just a few weeks ago, Zabrocki found a module extending the diagonal harmonics, whose bi-graded Frobenius characteristic he conjectured to be∆⁰_en−k−1e_n.

The Delta operators

Working on the Macdonald positivity conjecture, Garsia and Haiman introduced theS_N-moduleDH_nofdiagonal harmonics.

It turns out that

F(DH_n;q, t) =∇e_n

I ∇is the operator defined by∇He_µ:=T_µHe_µwhereT_µ∈N[q, t].

I TheDelta operators, for somef ∈Λare defined by

∆_fHe_µ:=f[B_µ(q, t)]He_µ and ∆⁰_fHe_µ:=f[B_µ(q, t)−1]He_µ, whereB_µ∈N[q, t].

I OnΛ⁽ⁿ⁾,

∆en =∇ and ∆ek = ∆⁰_ek+ ∆⁰_ek−1

I Just a few weeks ago, Zabrocki found a module extending the diagonal harmonics, whose bi-graded Frobenius characteristic he conjectured to be∆⁰_en−k−1e_n.

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

Combinatorial interpretations

Function Conjecture Proof

∇e_n= ∆enen

Shuffle conjecture Haglund, Haiman, Loehr

Remmel, Ulyanov, 2005.

Carlsson Mellit

2015

∆⁰_en−k−1en

Delta conjecture Haglund, Remmel,

Wilson, 2015

∆_hm∆⁰_en−k−1e_n

Generalised Delta conjecture

idem

∇(−1)ⁿ⁻¹p_n Square conjecture Loehr, Warrington, 2007

Sergel 2016

[n−k]t

[n]t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n

Generalised Delta square conjecture

D-I-VW

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

1 3

4 6

2 6

∗

∗

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

1 3

3 4 6

1 2 6

∗

∗

LD(n)^∗k: labelled decorated Dyck paths

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

3 3 4 6

1 2 6

∗

∗

LD(n)^∗k: labelled decorated Dyck paths

I Dyck path of sizen

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

1 3

3 4 6

1 2 6

∗

∗ LD(n)^∗k: labelled decorated Dyck paths

I Dyck path of sizen

I kdecorations onrises(i.e. vertical steps preceded by another

vertical step).

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

3 3 4 6

1 2 6

∗

∗

LD(n)^∗k: labelled decorated Dyck paths

I Dyck path of sizen

I kdecorations onrises(i.e. vertical steps preceded by another

vertical step).

I vertical steps labelled with

The Delta conjecture

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

1 3

3 4 6

1 2 6

∗

∗

LD(n)^∗k: labelled decorated Dyck paths

I Dyck path of sizen

I kdecorations onrises(i.e. vertical steps preceded by another

vertical step).

I vertical steps labelled with nonzero, positive integers

I labels strictly increasing in columns

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

3 0 4 6

0 2 6

∗

∗

PLD(m, n)^∗k: partiallylabelled decorated Dyck paths

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

1 3

0 4 6

0 2 6

∗

∗

PLD(m, n)^∗k: partiallylabelled decorated Dyck paths

I mzero labels,nnonzero labels I first label cannot be zero

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

3 0 4 6

0 2 6

∗

∗

Area: number of whole squares between the path andy=x, andnot in a row containing a decorated rise.

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

1 3 3

0 4 4 6

0 2 6

∗

∗

Dinv: count the number of pairs I same diagonal,

lower label < upper label (primary dinv)

I lower step one diagonal above upper step

lower label > upper label (secondary dinv)

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

3 3

0 0 4 6 6

0 0 2 6

∗

∗

Dinv: count the number of pairs I same diagonal,

lower label < upper label (primary dinv)

I lower step one diagonal above upper step

lower label > upper label

The generalised Delta conjecture

∆_hm∆⁰_en−k−1e_n= X

D∈PLD(m,n)^∗k

q^dinv(D)t^area(D)x^D

1 3

0 4 6

0 2 6

∗

∗

x^D :=

m+n

Y

i=1

x_li(D) wherel_i(D)is the label of thei-th vertical step ofDand we setx0 = 1.

Generalised Delta conjecture: state of the art

Conditions Reference

m= 0andk= 0 Carlsson-Mellit

m= 0andq = 0 Garsia-Haglund-Remmel-Yoo

m= 0andq = 1 Romero

m= 0andh·, h_n−dh_di D’Adderio-Iraci h·, e_n−dhdi D-I-VW

t= 0orq = 0 D-I-VW

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2

0 2 4

0 1 3

1

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

0 2 4

0 1 3

1

PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2

0 2 4

0 1 3

1

PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

I Square paths of sizem+nending east

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

0 2 4

0 1 3

1 PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

I Square paths of sizem+nending east

I mzero labels,nnonzero labels, strictly increasing in columns

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2

0 2 4

0 1 3

1 PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

I Square paths of sizem+nending east

I mzero labels,nnonzero labels, strictly increasing in columns I kdecorations on rises

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

0 2 4

0 1 3

1

PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

I Square paths of sizem+nending east

I mzero labels,nnonzero labels, strictly increasing in columns I kdecorations on rises

I At least one vertical step starting

The generalised Delta square conjecture

[n−k]_t

[n]_t ∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2

0 2 4

0 1 3

1

PLSQ^E(m, n)^∗k: partially labelled, decorated square paths ending east

I Square paths of sizem+nending east

I mzero labels,nnonzero labels, strictly increasing in columns I kdecorations on rises

I At least one vertical step starting from the lowest diagonal has a nonzero label

I if the first step is north, its label is nonzero.

The generalised Delta square conjecture

[n−k]t

[n]t

∆hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2 4

0 1 3

1

Area: number of whole squares between the path and the lowest diagonal touched by the path andnot in a row containing a decorated rise.

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2 2

0 0

2 2 4

0 1 3 3

1

Dinv

I Primary: same diagonal lower label < upper label

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

0 2 4

0 1 1 3

1 Dinv

I Primary: same diagonal lower label < upper label I Secondary: lower step one

diagonal above upper step lower label > upper label

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

2

0 2 2 4 4

0 1 1 1 3

1 Dinv

I Primary: same diagonal lower label < upper label I Secondary: lower step one

diagonal above upper step lower label > upper label

I Bonus: +1 for every nonzero label under the linex=y

The generalised Delta square conjecture

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P

∗

0 2 4

0 1 3

1

x^D :=

m+n

Y

i=1

x_li(D) whereli(D)is the label of thei-th vertical step ofDand we setx₀ = 1.

Support for our square conjecture

I k=m= 0is the square conjecture made by Loehr and Warrington, proven by Sergel.

I Computer evidence (using MAPLE and PYTHON) I We proved

I The caseq= 0. It coincides with the generalised Delta conjecture.

I The casek=t= 0, which is straightforward. I TheSchröder case, i.e.

[n−k]t

[n]t

h∆h_m∆en−k(−1)ⁿ⁻¹pn, en−dhdi

Support for our square conjecture

I k=m= 0is the square conjecture made by Loehr and Warrington, proven by Sergel.

I Computer evidence (using MAPLE and PYTHON)

I We proved

I The caseq= 0. It coincides with the generalised Delta conjecture.

I The casek=t= 0, which is straightforward. I TheSchröder case, i.e.

[n−k]t

[n]t

h∆h_m∆en−k(−1)ⁿ⁻¹pn, en−dhdi

Support for our square conjecture

I k=m= 0is the square conjecture made by Loehr and Warrington, proven by Sergel.

I Computer evidence (using MAPLE and PYTHON) I We proved

I The caseq= 0. It coincides with the generalised Delta conjecture.

I The casek=t= 0, which is straightforward. I TheSchröder case, i.e.

[n−k]t

[n]t

h∆h_m∆en−k(−1)ⁿ⁻¹pn, en−dhdi

Support for our square conjecture

I k=m= 0is the square conjecture made by Loehr and Warrington, proven by Sergel.

I Computer evidence (using MAPLE and PYTHON) I We proved

I The caseq= 0. It coincides with the generalised Delta conjecture.

I The casek=t= 0, which is straightforward.

I TheSchröder case, i.e.

[n−k]t

[n]t

h∆h_m∆en−k(−1)ⁿ⁻¹pn, en−dhdi

Support for our square conjecture

I k=m= 0is the square conjecture made by Loehr and Warrington, proven by Sergel.

I Computer evidence (using MAPLE and PYTHON) I We proved

I The caseq= 0. It coincides with the generalised Delta conjecture.

I The casek=t= 0, which is straightforward.

I TheSchröder case, i.e.

[n−k]t

[n]t

h∆h_m∆en−k(−1)ⁿ⁻¹pn, en−dhdi

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

0

0 The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 1 6

0 2 5

4

reading word

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 6

0 2 5

4 4

reading word 4

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 1 6

0 2 5 5

4

reading word 4 5

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 6

0 2 5

4

reading word 4 5 3

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 1 6

0 2 2 5

4

reading word 4 5 3 2

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 6

0 0 2 5

4

reading word 4 5 3 2 0

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 1 6 6

0 2 5

4

reading word 4 5 3 2 0 6

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 6

0 2 5

4

reading word 4 5 3 2 0 6 1

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 0

1 6

0 2 5

4

reading word 4 5 3 2 0 6 1 0

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dhdiof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 6

0 2 5

4 reading word

4 5 3 2 0 6 1 0 ...

4 5 3 2 0 6 1 0 4 5 3 2 0 6 1 0

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dhdiof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

3

0 1 6

0 2 5

4 reading word

4 5 3 2 0 6 1 0 ...

4 5 3 2 0 6 1 0 4 5 3 2 0 6 1 0 45 3 2 0 6 1 0

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]t

[n]_t ∆_hm∆en−k(−1)ⁿ⁻¹pn= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dhdiof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 0

reading word 4 5 3 2 0 6 1 0

...

4 5 3 2 0 6 1 0 4 5 3 2 0 6 1 0

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

0

0 SQ^E(m, n)^∗k,◦d

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗ 0

SQ^E(m, n)^∗k,◦d

I square paths of ending east of sizem+n

The steps labelledn−d+ 1, . . . , n must bepeaks.

Schröder case: combinatorial meaning

Suppose the generalised Delta conjecture is true, i.e.

[n−k]_t [n]t

∆_hm∆_en−k(−1)ⁿ⁻¹p_n= X

P∈PLSQ^E(m,n)^∗k

q^dinv(P)t^area(P)x^P.

Then takingh·, e_n−dh_diof this equation givesP

P∈Sq^dinv(D)t^area(P) on the RHS whereS⊆PLSQ^E(m, n)^∗kis the set of paths whose reading wordis ashuffleofmzeroes, the stringn−d, . . . ,1and the stringn−d+ 1, . . . , n.

∗

0 0

SQ^E(m, n)^∗k,◦d

I square paths of ending east of sizem+n

I mzero labels in valleys

The steps labelledn−d+ 1, . . . , n must bepeaks.