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... => 4
[1,0,1,1,0,0] generating graphics... => 6
[1,1,0,0,1,0] generating graphics... => 6
[1,1,0,1,0,0] generating graphics... => 9
[1,1,1,0,0,0] generating graphics... => 15
[1,0,1,0,1,0,1,0] generating graphics... => 10
[1,0,1,0,1,1,0,0] generating graphics... => 15
[1,0,1,1,0,0,1,0] generating graphics... => 15
[1,0,1,1,0,1,0,0] generating graphics... => 22
[1,0,1,1,1,0,0,0] generating graphics... => 36
[1,1,0,0,1,0,1,0] generating graphics... => 15
[1,1,0,0,1,1,0,0] generating graphics... => 23
[1,1,0,1,0,0,1,0] generating graphics... => 22
[1,1,0,1,0,1,0,0] generating graphics... => 33
[1,1,0,1,1,0,0,0] generating graphics... => 53
[1,1,1,0,0,0,1,0] generating graphics... => 36
[1,1,1,0,0,1,0,0] generating graphics... => 53
[1,1,1,0,1,0,0,0] generating graphics... => 87
[1,1,1,1,0,0,0,0] generating graphics... => 155
[1,0,1,0,1,0,1,0,1,0] generating graphics... => 26
[1,0,1,0,1,0,1,1,0,0] generating graphics... => 39
[1,0,1,0,1,1,0,0,1,0] generating graphics... => 39
[1,0,1,0,1,1,0,1,0,0] generating graphics... => 57
[1,0,1,0,1,1,1,0,0,0] generating graphics... => 93
[1,0,1,1,0,0,1,0,1,0] generating graphics... => 39
[1,0,1,1,0,0,1,1,0,0] generating graphics... => 59
[1,0,1,1,0,1,0,0,1,0] generating graphics... => 57
[1,0,1,1,0,1,0,1,0,0] generating graphics... => 84
[1,0,1,1,0,1,1,0,0,0] generating graphics... => 134
[1,0,1,1,1,0,0,0,1,0] generating graphics... => 93
[1,0,1,1,1,0,0,1,0,0] generating graphics... => 134
[1,0,1,1,1,0,1,0,0,0] generating graphics... => 216
[1,0,1,1,1,1,0,0,0,0] generating graphics... => 380
[1,1,0,0,1,0,1,0,1,0] generating graphics... => 39
[1,1,0,0,1,0,1,1,0,0] generating graphics... => 59
[1,1,0,0,1,1,0,0,1,0] generating graphics... => 59
[1,1,0,0,1,1,0,1,0,0] generating graphics... => 87
[1,1,0,0,1,1,1,0,0,0] generating graphics... => 143
[1,1,0,1,0,0,1,0,1,0] generating graphics... => 57
[1,1,0,1,0,0,1,1,0,0] generating graphics... => 87
[1,1,0,1,0,1,0,0,1,0] generating graphics... => 84
[1,1,0,1,0,1,0,1,0,0] generating graphics... => 125
[1,1,0,1,0,1,1,0,0,0] generating graphics... => 201
[1,1,0,1,1,0,0,0,1,0] generating graphics... => 134
[1,1,0,1,1,0,0,1,0,0] generating graphics... => 195
[1,1,0,1,1,0,1,0,0,0] generating graphics... => 317
[1,1,0,1,1,1,0,0,0,0] generating graphics... => 549
[1,1,1,0,0,0,1,0,1,0] generating graphics... => 93
[1,1,1,0,0,0,1,1,0,0] generating graphics... => 143
[1,1,1,0,0,1,0,0,1,0] generating graphics... => 134
[1,1,1,0,0,1,0,1,0,0] generating graphics... => 201
[1,1,1,0,0,1,1,0,0,0] generating graphics... => 317
[1,1,1,0,1,0,0,0,1,0] generating graphics... => 216
[1,1,1,0,1,0,0,1,0,0] generating graphics... => 317
[1,1,1,0,1,0,1,0,0,0] generating graphics... => 507
[1,1,1,0,1,1,0,0,0,0] generating graphics... => 887
[1,1,1,1,0,0,0,0,1,0] generating graphics... => 380
[1,1,1,1,0,0,0,1,0,0] generating graphics... => 549
[1,1,1,1,0,0,1,0,0,0] generating graphics... => 887
[1,1,1,1,0,1,0,0,0,0] generating graphics... => 1563
[1,1,1,1,1,0,0,0,0,0] generating graphics... => 2915
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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.
References
[1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles arXiv:1705.10353
Created
Sep 05, 2018 at 08:58 by Per Alexandersson
Updated
Sep 05, 2018 at 08:58 by Per Alexandersson