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Statistic identifier: St001243
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Collection: Dyck paths
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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.
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References: [1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles [[arXiv:1705.10353]]
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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] => 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
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Created: Sep 05, 2018 at 08:58 by Per Alexandersson
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Last Updated: Sep 05, 2018 at 08:58 by Per Alexandersson