Identifier
Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>1 [1,0,1,0]=>2 [1,1,0,0]=>3 [1,0,1,0,1,0]=>6 [1,0,1,1,0,0]=>9 [1,1,0,0,1,0]=>9 [1,1,0,1,0,0]=>13 [1,1,1,0,0,0]=>21 [1,0,1,0,1,0,1,0]=>24 [1,0,1,0,1,1,0,0]=>36 [1,0,1,1,0,0,1,0]=>36 [1,0,1,1,0,1,0,0]=>52 [1,0,1,1,1,0,0,0]=>84 [1,1,0,0,1,0,1,0]=>36 [1,1,0,0,1,1,0,0]=>54 [1,1,0,1,0,0,1,0]=>52 [1,1,0,1,0,1,0,0]=>75 [1,1,0,1,1,0,0,0]=>117 [1,1,1,0,0,0,1,0]=>84 [1,1,1,0,0,1,0,0]=>117 [1,1,1,0,1,0,0,0]=>183 [1,1,1,1,0,0,0,0]=>315 [1,0,1,0,1,0,1,0,1,0]=>120 [1,0,1,0,1,0,1,1,0,0]=>180 [1,0,1,0,1,1,0,0,1,0]=>180 [1,0,1,0,1,1,0,1,0,0]=>260 [1,0,1,0,1,1,1,0,0,0]=>420 [1,0,1,1,0,0,1,0,1,0]=>180 [1,0,1,1,0,0,1,1,0,0]=>270 [1,0,1,1,0,1,0,0,1,0]=>260 [1,0,1,1,0,1,0,1,0,0]=>375 [1,0,1,1,0,1,1,0,0,0]=>585 [1,0,1,1,1,0,0,0,1,0]=>420 [1,0,1,1,1,0,0,1,0,0]=>585 [1,0,1,1,1,0,1,0,0,0]=>915 [1,0,1,1,1,1,0,0,0,0]=>1575 [1,1,0,0,1,0,1,0,1,0]=>180 [1,1,0,0,1,0,1,1,0,0]=>270 [1,1,0,0,1,1,0,0,1,0]=>270 [1,1,0,0,1,1,0,1,0,0]=>390 [1,1,0,0,1,1,1,0,0,0]=>630 [1,1,0,1,0,0,1,0,1,0]=>260 [1,1,0,1,0,0,1,1,0,0]=>390 [1,1,0,1,0,1,0,0,1,0]=>375 [1,1,0,1,0,1,0,1,0,0]=>541 [1,1,0,1,0,1,1,0,0,0]=>843 [1,1,0,1,1,0,0,0,1,0]=>585 [1,1,0,1,1,0,0,1,0,0]=>813 [1,1,0,1,1,0,1,0,0,0]=>1269 [1,1,0,1,1,1,0,0,0,0]=>2121 [1,1,1,0,0,0,1,0,1,0]=>420 [1,1,1,0,0,0,1,1,0,0]=>630 [1,1,1,0,0,1,0,0,1,0]=>585 [1,1,1,0,0,1,0,1,0,0]=>843 [1,1,1,0,0,1,1,0,0,0]=>1269 [1,1,1,0,1,0,0,0,1,0]=>915 [1,1,1,0,1,0,0,1,0,0]=>1269 [1,1,1,0,1,0,1,0,0,0]=>1917 [1,1,1,0,1,1,0,0,0,0]=>3213 [1,1,1,1,0,0,0,0,1,0]=>1575 [1,1,1,1,0,0,0,1,0,0]=>2121 [1,1,1,1,0,0,1,0,0,0]=>3213 [1,1,1,1,0,1,0,0,0,0]=>5397 [1,1,1,1,1,0,0,0,0,0]=>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