- FindStat

Identifier

St001243: Dyck paths ⟶ ℤ

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
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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