Identifier
Identifier
Values
[1,0] generating graphics... => 1
[1,0,1,0] generating graphics... => 2
[1,1,0,0] generating graphics... => 3
[1,0,1,0,1,0] generating graphics... => 6
[1,0,1,1,0,0] generating graphics... => 9
[1,1,0,0,1,0] generating graphics... => 9
[1,1,0,1,0,0] generating graphics... => 13
[1,1,1,0,0,0] generating graphics... => 21
[1,0,1,0,1,0,1,0] generating graphics... => 24
[1,0,1,0,1,1,0,0] generating graphics... => 36
[1,0,1,1,0,0,1,0] generating graphics... => 36
[1,0,1,1,0,1,0,0] generating graphics... => 52
[1,0,1,1,1,0,0,0] generating graphics... => 84
[1,1,0,0,1,0,1,0] generating graphics... => 36
[1,1,0,0,1,1,0,0] generating graphics... => 54
[1,1,0,1,0,0,1,0] generating graphics... => 52
[1,1,0,1,0,1,0,0] generating graphics... => 75
[1,1,0,1,1,0,0,0] generating graphics... => 117
[1,1,1,0,0,0,1,0] generating graphics... => 84
[1,1,1,0,0,1,0,0] generating graphics... => 117
[1,1,1,0,1,0,0,0] generating graphics... => 183
[1,1,1,1,0,0,0,0] generating graphics... => 315
[1,0,1,0,1,0,1,0,1,0] generating graphics... => 120
[1,0,1,0,1,0,1,1,0,0] generating graphics... => 180
[1,0,1,0,1,1,0,0,1,0] generating graphics... => 180
[1,0,1,0,1,1,0,1,0,0] generating graphics... => 260
[1,0,1,0,1,1,1,0,0,0] generating graphics... => 420
[1,0,1,1,0,0,1,0,1,0] generating graphics... => 180
[1,0,1,1,0,0,1,1,0,0] generating graphics... => 270
[1,0,1,1,0,1,0,0,1,0] generating graphics... => 260
[1,0,1,1,0,1,0,1,0,0] generating graphics... => 375
[1,0,1,1,0,1,1,0,0,0] generating graphics... => 585
[1,0,1,1,1,0,0,0,1,0] generating graphics... => 420
[1,0,1,1,1,0,0,1,0,0] generating graphics... => 585
[1,0,1,1,1,0,1,0,0,0] generating graphics... => 915
[1,0,1,1,1,1,0,0,0,0] generating graphics... => 1575
[1,1,0,0,1,0,1,0,1,0] generating graphics... => 180
[1,1,0,0,1,0,1,1,0,0] generating graphics... => 270
[1,1,0,0,1,1,0,0,1,0] generating graphics... => 270
[1,1,0,0,1,1,0,1,0,0] generating graphics... => 390
[1,1,0,0,1,1,1,0,0,0] generating graphics... => 630
[1,1,0,1,0,0,1,0,1,0] generating graphics... => 260
[1,1,0,1,0,0,1,1,0,0] generating graphics... => 390
[1,1,0,1,0,1,0,0,1,0] generating graphics... => 375
[1,1,0,1,0,1,0,1,0,0] generating graphics... => 541
[1,1,0,1,0,1,1,0,0,0] generating graphics... => 843
[1,1,0,1,1,0,0,0,1,0] generating graphics... => 585
[1,1,0,1,1,0,0,1,0,0] generating graphics... => 813
[1,1,0,1,1,0,1,0,0,0] generating graphics... => 1269
[1,1,0,1,1,1,0,0,0,0] generating graphics... => 2121
[1,1,1,0,0,0,1,0,1,0] generating graphics... => 420
[1,1,1,0,0,0,1,1,0,0] generating graphics... => 630
[1,1,1,0,0,1,0,0,1,0] generating graphics... => 585
[1,1,1,0,0,1,0,1,0,0] generating graphics... => 843
[1,1,1,0,0,1,1,0,0,0] generating graphics... => 1269
[1,1,1,0,1,0,0,0,1,0] generating graphics... => 915
[1,1,1,0,1,0,0,1,0,0] generating graphics... => 1269
[1,1,1,0,1,0,1,0,0,0] generating graphics... => 1917
[1,1,1,0,1,1,0,0,0,0] generating graphics... => 3213
[1,1,1,1,0,0,0,0,1,0] generating graphics... => 1575
[1,1,1,1,0,0,0,1,0,0] generating graphics... => 2121
[1,1,1,1,0,0,1,0,0,0] generating graphics... => 3213
[1,1,1,1,0,1,0,0,0,0] generating graphics... => 5397
[1,1,1,1,1,0,0,0,0,0] generating graphics... => 9765
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Description
The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path.
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.
Thus, $G_\Gamma(x;q+1)$ is the Frobenius image of some (graded) $S_n$-module.
The total dimension of this $S_n$-module is
$$D_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1)f^\lambda$$
where $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$.
This statistic is $D_\Gamma$.
References
[1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles DOI:10.1016/j.disc.2018.09.001
Created
Sep 05, 2018 at 08:45 by Per Alexandersson
Updated
Sep 25, 2018 at 13:07 by Per Alexandersson