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Identifier
Values
=>
Cc0005;cc-rep
[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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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