***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000006 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The dinv of a Dyck path. Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]). The dinv statistic of $D$ is $$\operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$ Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by $$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$ ----------------------------------------------------------------------------- References: [1] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics [[MathSciNet:2371044]] [2] Garsia, A. M., Xin, G., Zabrocki, M. Hall-Littlewood operators in the theory of parking functions and diagonal harmonics [[MathSciNet:2899952]] ----------------------------------------------------------------------------- Code: def statistic(x): return x.dinv() ----------------------------------------------------------------------------- Statistic values: [1,0] => 0 [1,0,1,0] => 1 [1,1,0,0] => 0 [1,0,1,0,1,0] => 3 [1,0,1,1,0,0] => 1 [1,1,0,0,1,0] => 2 [1,1,0,1,0,0] => 1 [1,1,1,0,0,0] => 0 [1,0,1,0,1,0,1,0] => 6 [1,0,1,0,1,1,0,0] => 3 [1,0,1,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,0] => 2 [1,0,1,1,1,0,0,0] => 1 [1,1,0,0,1,0,1,0] => 5 [1,1,0,0,1,1,0,0] => 3 [1,1,0,1,0,0,1,0] => 4 [1,1,0,1,0,1,0,0] => 3 [1,1,0,1,1,0,0,0] => 1 [1,1,1,0,0,0,1,0] => 2 [1,1,1,0,0,1,0,0] => 2 [1,1,1,0,1,0,0,0] => 1 [1,1,1,1,0,0,0,0] => 0 [1,0,1,0,1,0,1,0,1,0] => 10 [1,0,1,0,1,0,1,1,0,0] => 6 [1,0,1,0,1,1,0,0,1,0] => 7 [1,0,1,0,1,1,0,1,0,0] => 4 [1,0,1,0,1,1,1,0,0,0] => 3 [1,0,1,1,0,0,1,0,1,0] => 8 [1,0,1,1,0,0,1,1,0,0] => 5 [1,0,1,1,0,1,0,0,1,0] => 6 [1,0,1,1,0,1,0,1,0,0] => 4 [1,0,1,1,0,1,1,0,0,0] => 2 [1,0,1,1,1,0,0,0,1,0] => 4 [1,0,1,1,1,0,0,1,0,0] => 3 [1,0,1,1,1,0,1,0,0,0] => 2 [1,0,1,1,1,1,0,0,0,0] => 1 [1,1,0,0,1,0,1,0,1,0] => 9 [1,1,0,0,1,0,1,1,0,0] => 6 [1,1,0,0,1,1,0,0,1,0] => 7 [1,1,0,0,1,1,0,1,0,0] => 5 [1,1,0,0,1,1,1,0,0,0] => 3 [1,1,0,1,0,0,1,0,1,0] => 8 [1,1,0,1,0,0,1,1,0,0] => 6 [1,1,0,1,0,1,0,0,1,0] => 7 [1,1,0,1,0,1,0,1,0,0] => 6 [1,1,0,1,0,1,1,0,0,0] => 3 [1,1,0,1,1,0,0,0,1,0] => 4 [1,1,0,1,1,0,0,1,0,0] => 4 [1,1,0,1,1,0,1,0,0,0] => 2 [1,1,0,1,1,1,0,0,0,0] => 1 [1,1,1,0,0,0,1,0,1,0] => 5 [1,1,1,0,0,0,1,1,0,0] => 4 [1,1,1,0,0,1,0,0,1,0] => 5 [1,1,1,0,0,1,0,1,0,0] => 5 [1,1,1,0,0,1,1,0,0,0] => 3 [1,1,1,0,1,0,0,0,1,0] => 3 [1,1,1,0,1,0,0,1,0,0] => 4 [1,1,1,0,1,0,1,0,0,0] => 3 [1,1,1,0,1,1,0,0,0,0] => 1 [1,1,1,1,0,0,0,0,1,0] => 2 [1,1,1,1,0,0,0,1,0,0] => 2 [1,1,1,1,0,0,1,0,0,0] => 2 [1,1,1,1,0,1,0,0,0,0] => 1 [1,1,1,1,1,0,0,0,0,0] => 0 [1,0,1,0,1,0,1,0,1,0,1,0] => 15 [1,0,1,0,1,0,1,0,1,1,0,0] => 10 [1,0,1,0,1,0,1,1,0,0,1,0] => 11 [1,0,1,0,1,0,1,1,0,1,0,0] => 7 [1,0,1,0,1,0,1,1,1,0,0,0] => 6 [1,0,1,0,1,1,0,0,1,0,1,0] => 12 [1,0,1,0,1,1,0,0,1,1,0,0] => 8 [1,0,1,0,1,1,0,1,0,0,1,0] => 9 [1,0,1,0,1,1,0,1,0,1,0,0] => 6 [1,0,1,0,1,1,0,1,1,0,0,0] => 4 [1,0,1,0,1,1,1,0,0,0,1,0] => 7 [1,0,1,0,1,1,1,0,0,1,0,0] => 5 [1,0,1,0,1,1,1,0,1,0,0,0] => 4 [1,0,1,0,1,1,1,1,0,0,0,0] => 3 [1,0,1,1,0,0,1,0,1,0,1,0] => 13 [1,0,1,1,0,0,1,0,1,1,0,0] => 9 [1,0,1,1,0,0,1,1,0,0,1,0] => 10 [1,0,1,1,0,0,1,1,0,1,0,0] => 7 [1,0,1,1,0,0,1,1,1,0,0,0] => 5 [1,0,1,1,0,1,0,0,1,0,1,0] => 11 [1,0,1,1,0,1,0,0,1,1,0,0] => 8 [1,0,1,1,0,1,0,1,0,0,1,0] => 9 [1,0,1,1,0,1,0,1,0,1,0,0] => 7 [1,0,1,1,0,1,0,1,1,0,0,0] => 4 [1,0,1,1,0,1,1,0,0,0,1,0] => 6 [1,0,1,1,0,1,1,0,0,1,0,0] => 5 [1,0,1,1,0,1,1,0,1,0,0,0] => 3 [1,0,1,1,0,1,1,1,0,0,0,0] => 2 [1,0,1,1,1,0,0,0,1,0,1,0] => 8 [1,0,1,1,1,0,0,0,1,1,0,0] => 6 [1,0,1,1,1,0,0,1,0,0,1,0] => 7 [1,0,1,1,1,0,0,1,0,1,0,0] => 6 [1,0,1,1,1,0,0,1,1,0,0,0] => 4 [1,0,1,1,1,0,1,0,0,0,1,0] => 5 [1,0,1,1,1,0,1,0,0,1,0,0] => 5 [1,0,1,1,1,0,1,0,1,0,0,0] => 4 [1,0,1,1,1,0,1,1,0,0,0,0] => 2 [1,0,1,1,1,1,0,0,0,0,1,0] => 4 [1,0,1,1,1,1,0,0,0,1,0,0] => 3 [1,0,1,1,1,1,0,0,1,0,0,0] => 3 [1,0,1,1,1,1,0,1,0,0,0,0] => 2 [1,0,1,1,1,1,1,0,0,0,0,0] => 1 [1,1,0,0,1,0,1,0,1,0,1,0] => 14 [1,1,0,0,1,0,1,0,1,1,0,0] => 10 [1,1,0,0,1,0,1,1,0,0,1,0] => 11 [1,1,0,0,1,0,1,1,0,1,0,0] => 8 [1,1,0,0,1,0,1,1,1,0,0,0] => 6 [1,1,0,0,1,1,0,0,1,0,1,0] => 12 [1,1,0,0,1,1,0,0,1,1,0,0] => 9 [1,1,0,0,1,1,0,1,0,0,1,0] => 10 [1,1,0,0,1,1,0,1,0,1,0,0] => 8 [1,1,0,0,1,1,0,1,1,0,0,0] => 5 [1,1,0,0,1,1,1,0,0,0,1,0] => 7 [1,1,0,0,1,1,1,0,0,1,0,0] => 6 [1,1,0,0,1,1,1,0,1,0,0,0] => 4 [1,1,0,0,1,1,1,1,0,0,0,0] => 3 [1,1,0,1,0,0,1,0,1,0,1,0] => 13 [1,1,0,1,0,0,1,0,1,1,0,0] => 10 [1,1,0,1,0,0,1,1,0,0,1,0] => 11 [1,1,0,1,0,0,1,1,0,1,0,0] => 9 [1,1,0,1,0,0,1,1,1,0,0,0] => 6 [1,1,0,1,0,1,0,0,1,0,1,0] => 12 [1,1,0,1,0,1,0,0,1,1,0,0] => 10 [1,1,0,1,0,1,0,1,0,0,1,0] => 11 [1,1,0,1,0,1,0,1,0,1,0,0] => 10 [1,1,0,1,0,1,0,1,1,0,0,0] => 6 [1,1,0,1,0,1,1,0,0,0,1,0] => 7 [1,1,0,1,0,1,1,0,0,1,0,0] => 7 [1,1,0,1,0,1,1,0,1,0,0,0] => 4 [1,1,0,1,0,1,1,1,0,0,0,0] => 3 [1,1,0,1,1,0,0,0,1,0,1,0] => 8 [1,1,0,1,1,0,0,0,1,1,0,0] => 7 [1,1,0,1,1,0,0,1,0,0,1,0] => 8 [1,1,0,1,1,0,0,1,0,1,0,0] => 8 [1,1,0,1,1,0,0,1,1,0,0,0] => 5 [1,1,0,1,1,0,1,0,0,0,1,0] => 5 [1,1,0,1,1,0,1,0,0,1,0,0] => 6 [1,1,0,1,1,0,1,0,1,0,0,0] => 4 [1,1,0,1,1,0,1,1,0,0,0,0] => 2 [1,1,0,1,1,1,0,0,0,0,1,0] => 4 [1,1,0,1,1,1,0,0,0,1,0,0] => 4 [1,1,0,1,1,1,0,0,1,0,0,0] => 3 [1,1,0,1,1,1,0,1,0,0,0,0] => 2 [1,1,0,1,1,1,1,0,0,0,0,0] => 1 [1,1,1,0,0,0,1,0,1,0,1,0] => 9 [1,1,1,0,0,0,1,0,1,1,0,0] => 7 [1,1,1,0,0,0,1,1,0,0,1,0] => 8 [1,1,1,0,0,0,1,1,0,1,0,0] => 7 [1,1,1,0,0,0,1,1,1,0,0,0] => 5 [1,1,1,0,0,1,0,0,1,0,1,0] => 9 [1,1,1,0,0,1,0,0,1,1,0,0] => 8 [1,1,1,0,0,1,0,1,0,0,1,0] => 9 [1,1,1,0,0,1,0,1,0,1,0,0] => 9 [1,1,1,0,0,1,0,1,1,0,0,0] => 6 [1,1,1,0,0,1,1,0,0,0,1,0] => 6 [1,1,1,0,0,1,1,0,0,1,0,0] => 7 [1,1,1,0,0,1,1,0,1,0,0,0] => 5 [1,1,1,0,0,1,1,1,0,0,0,0] => 3 [1,1,1,0,1,0,0,0,1,0,1,0] => 6 [1,1,1,0,1,0,0,0,1,1,0,0] => 6 [1,1,1,0,1,0,0,1,0,0,1,0] => 7 [1,1,1,0,1,0,0,1,0,1,0,0] => 8 [1,1,1,0,1,0,0,1,1,0,0,0] => 6 [1,1,1,0,1,0,1,0,0,0,1,0] => 5 [1,1,1,0,1,0,1,0,0,1,0,0] => 7 [1,1,1,0,1,0,1,0,1,0,0,0] => 6 [1,1,1,0,1,0,1,1,0,0,0,0] => 3 [1,1,1,0,1,1,0,0,0,0,1,0] => 3 [1,1,1,0,1,1,0,0,0,1,0,0] => 4 [1,1,1,0,1,1,0,0,1,0,0,0] => 4 [1,1,1,0,1,1,0,1,0,0,0,0] => 2 [1,1,1,0,1,1,1,0,0,0,0,0] => 1 [1,1,1,1,0,0,0,0,1,0,1,0] => 5 [1,1,1,1,0,0,0,0,1,1,0,0] => 4 [1,1,1,1,0,0,0,1,0,0,1,0] => 5 [1,1,1,1,0,0,0,1,0,1,0,0] => 5 [1,1,1,1,0,0,0,1,1,0,0,0] => 4 [1,1,1,1,0,0,1,0,0,0,1,0] => 4 [1,1,1,1,0,0,1,0,0,1,0,0] => 5 [1,1,1,1,0,0,1,0,1,0,0,0] => 5 [1,1,1,1,0,0,1,1,0,0,0,0] => 3 [1,1,1,1,0,1,0,0,0,0,1,0] => 3 [1,1,1,1,0,1,0,0,0,1,0,0] => 3 [1,1,1,1,0,1,0,0,1,0,0,0] => 4 [1,1,1,1,0,1,0,1,0,0,0,0] => 3 [1,1,1,1,0,1,1,0,0,0,0,0] => 1 [1,1,1,1,1,0,0,0,0,0,1,0] => 2 [1,1,1,1,1,0,0,0,0,1,0,0] => 2 [1,1,1,1,1,0,0,0,1,0,0,0] => 2 [1,1,1,1,1,0,0,1,0,0,0,0] => 2 [1,1,1,1,1,0,1,0,0,0,0,0] => 1 [1,1,1,1,1,1,0,0,0,0,0,0] => 0 ----------------------------------------------------------------------------- Created: Sep 21, 2011 at 03:34 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Jun 17, 2019 at 17:06 by Christian Stump