- FindStat

Identifier

St001243: Dyck paths ⟶ ℤ

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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$F_{1} = q$

$F_{2} = q^{2} + q^{3}$

$F_{3} = q^{4} + 2\ q^{6} + q^{9} + q^{15}$

$F_{4} = q^{10} + 3\ q^{15} + 2\ q^{22} + q^{23} + q^{33} + 2\ q^{36} + 2\ q^{53} + q^{87} + q^{155}$

$F_{5} = q^{26} + 4\ q^{39} + 3\ q^{57} + 3\ q^{59} + 2\ q^{84} + 2\ q^{87} + 3\ q^{93} + q^{125} + 4\ q^{134} + 2\ q^{143} + q^{195} + 2\ q^{201} + 2\ q^{216} + 3\ q^{317} + 2\ q^{380} + q^{507} + 2\ q^{549} + 2\ q^{887} + q^{1563} + q^{2915}$

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