**Identifier**

Identifier

- St001243: Dyck paths ⟶ ℤ

Values

[1,0]
=>
1

[1,0,1,0]
=>
2

[1,1,0,0]
=>
3

[1,0,1,0,1,0]
=>
4

[1,0,1,1,0,0]
=>
6

[1,1,0,0,1,0]
=>
6

[1,1,0,1,0,0]
=>
9

[1,1,1,0,0,0]
=>
15

[1,0,1,0,1,0,1,0]
=>
10

[1,0,1,0,1,1,0,0]
=>
15

[1,0,1,1,0,0,1,0]
=>
15

[1,0,1,1,0,1,0,0]
=>
22

[1,0,1,1,1,0,0,0]
=>
36

[1,1,0,0,1,0,1,0]
=>
15

[1,1,0,0,1,1,0,0]
=>
23

[1,1,0,1,0,0,1,0]
=>
22

[1,1,0,1,0,1,0,0]
=>
33

[1,1,0,1,1,0,0,0]
=>
53

[1,1,1,0,0,0,1,0]
=>
36

[1,1,1,0,0,1,0,0]
=>
53

[1,1,1,0,1,0,0,0]
=>
87

[1,1,1,1,0,0,0,0]
=>
155

[1,0,1,0,1,0,1,0,1,0]
=>
26

[1,0,1,0,1,0,1,1,0,0]
=>
39

[1,0,1,0,1,1,0,0,1,0]
=>
39

[1,0,1,0,1,1,0,1,0,0]
=>
57

[1,0,1,0,1,1,1,0,0,0]
=>
93

[1,0,1,1,0,0,1,0,1,0]
=>
39

[1,0,1,1,0,0,1,1,0,0]
=>
59

[1,0,1,1,0,1,0,0,1,0]
=>
57

[1,0,1,1,0,1,0,1,0,0]
=>
84

[1,0,1,1,0,1,1,0,0,0]
=>
134

[1,0,1,1,1,0,0,0,1,0]
=>
93

[1,0,1,1,1,0,0,1,0,0]
=>
134

[1,0,1,1,1,0,1,0,0,0]
=>
216

[1,0,1,1,1,1,0,0,0,0]
=>
380

[1,1,0,0,1,0,1,0,1,0]
=>
39

[1,1,0,0,1,0,1,1,0,0]
=>
59

[1,1,0,0,1,1,0,0,1,0]
=>
59

[1,1,0,0,1,1,0,1,0,0]
=>
87

[1,1,0,0,1,1,1,0,0,0]
=>
143

[1,1,0,1,0,0,1,0,1,0]
=>
57

[1,1,0,1,0,0,1,1,0,0]
=>
87

[1,1,0,1,0,1,0,0,1,0]
=>
84

[1,1,0,1,0,1,0,1,0,0]
=>
125

[1,1,0,1,0,1,1,0,0,0]
=>
201

[1,1,0,1,1,0,0,0,1,0]
=>
134

[1,1,0,1,1,0,0,1,0,0]
=>
195

[1,1,0,1,1,0,1,0,0,0]
=>
317

[1,1,0,1,1,1,0,0,0,0]
=>
549

[1,1,1,0,0,0,1,0,1,0]
=>
93

[1,1,1,0,0,0,1,1,0,0]
=>
143

[1,1,1,0,0,1,0,0,1,0]
=>
134

[1,1,1,0,0,1,0,1,0,0]
=>
201

[1,1,1,0,0,1,1,0,0,0]
=>
317

[1,1,1,0,1,0,0,0,1,0]
=>
216

[1,1,1,0,1,0,0,1,0,0]
=>
317

[1,1,1,0,1,0,1,0,0,0]
=>
507

[1,1,1,0,1,1,0,0,0,0]
=>
887

[1,1,1,1,0,0,0,0,1,0]
=>
380

[1,1,1,1,0,0,0,1,0,0]
=>
549

[1,1,1,1,0,0,1,0,0,0]
=>
887

[1,1,1,1,0,1,0,0,0,0]
=>
1563

[1,1,1,1,1,0,0,0,0,0]
=>
2915

Description

The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path.

In other words, given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$.

Consider the expansion

$$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$

using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$

is a so called unicellular LLT polynomial, and a symmetric function.

Consider the Schur expansion

$$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$

By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients.

Consider the sum

$$S_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1).$$

This statistic is $S_\Gamma$.

It is still an open problem to find a combinatorial description of the above Schur expansion,

a first step would be to find a family of combinatorial objects to sum over.

In other words, given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$.

Consider the expansion

$$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$

using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$

is a so called unicellular LLT polynomial, and a symmetric function.

Consider the Schur expansion

$$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$

By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients.

Consider the sum

$$S_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1).$$

This statistic is $S_\Gamma$.

It is still an open problem to find a combinatorial description of the above Schur expansion,

a first step would be to find a family of combinatorial objects to sum over.

References

[1]

**Alexandersson, P., Panova, G.***LLT polynomials, chromatic quasisymmetric functions and graphs with cycles*arXiv:1705.10353Created

Sep 05, 2018 at 08:58 by

**Per Alexandersson**Updated

Sep 05, 2018 at 08:58 by

**Per Alexandersson**searching the database

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