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Statistic identifier: St001242
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Collection: Dyck paths
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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 $D_\Gamma$.
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References: [1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles [[DOI:10.1016/j.disc.2018.09.001]]
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Code:
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Statistic values:
[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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Created: Sep 05, 2018 at 08:45 by Per Alexandersson
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Last Updated: Sep 25, 2018 at 13:07 by Per Alexandersson