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"StatisticIdentifier" : 1243,
"StatisticCollection" : "Dyck paths",
"StatisticDescription" : "The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path.\r\n\r\nIn other words, given a Dyck path, there is an associated (directed) unit interval graph $\\Gamma$.\r\n\r\nConsider the expansion\r\n$$G_\\Gamma(x;q) = \\sum_{\\kappa: V(G) \\to \\mathbb{N}_+} x_\\kappa q^{\\mathrm{asc}(\\kappa)}$$\r\nusing the notation by Alexandersson and Panova. The function $G_\\Gamma(x;q)$\r\nis a so called unicellular LLT polynomial, and a symmetric function.\r\n\r\nConsider the Schur expansion\r\n$$G_\\Gamma(x;q+1) = \\sum_{\\lambda} c^\\Gamma_\\lambda(q) s_\\lambda(x).$$\r\nBy a result by Haiman and Grojnowski, all $c^\\Gamma_\\lambda(q)$ have non-negative integer coefficients.\r\n\r\nConsider the sum\r\n$$S_\\Gamma = \\sum_{\\lambda} c^\\Gamma_\\lambda(1).$$\r\n\r\nThis statistic is $S_\\Gamma$. \r\n\r\nIt is still an open problem to find a combinatorial description of the above Schur expansion,\r\na first step would be to find a family of combinatorial objects to sum over.\r\n",
"StatisticTitle" : "The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path.",
"StatisticReferences" : "[1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles [[arXiv:1705.10353]]",
"StatisticCode" : "",
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"StatisticOriginalAuthor" : {"Name": "Per Alexandersson", "Time": "2018-09-05T08:58:33Z"},
"StatisticUpdateAuthor" : {"Name": "Per Alexandersson", "Time": "2018-09-05T08:58:33Z"},
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