***************************************************************************** * 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: St000005 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The bounce statistic of a Dyck path. The '''bounce path''' \$D'\$ of a Dyck path \$D\$ is the Dyck path obtained from \$D\$ by starting at the end point \$(2n,0)\$, traveling north-west until hitting \$D\$, then bouncing back south-west to the \$x\$-axis, and repeating this procedure until finally reaching the point \$(0,0)\$. The points where \$D'\$ touches the \$x\$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all \$i\$ for which the bounce path \$D'\$ of \$D\$ touches the \$x\$-axis at \$(2i,0)\$. In particular, the bounce statistics of \$D\$ and \$D'\$ coincide. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(x): return x.bounce() ----------------------------------------------------------------------------- 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] => 3 [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] => 2 [1,1,0,1,0,0,1,0] => 4 [1,1,0,1,0,1,0,0] => 2 [1,1,0,1,1,0,0,0] => 1 [1,1,1,0,0,0,1,0] => 3 [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] => 6 [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] => 4 [1,0,1,1,0,1,0,0,1,0] => 7 [1,0,1,1,0,1,0,1,0,0] => 4 [1,0,1,1,0,1,1,0,0,0] => 3 [1,0,1,1,1,0,0,0,1,0] => 5 [1,0,1,1,1,0,0,1,0,0] => 4 [1,0,1,1,1,0,1,0,0,0] => 3 [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] => 5 [1,1,0,0,1,1,0,0,1,0] => 6 [1,1,0,0,1,1,0,1,0,0] => 5 [1,1,0,0,1,1,1,0,0,0] => 2 [1,1,0,1,0,0,1,0,1,0] => 8 [1,1,0,1,0,0,1,1,0,0] => 4 [1,1,0,1,0,1,0,0,1,0] => 6 [1,1,0,1,0,1,0,1,0,0] => 4 [1,1,0,1,0,1,1,0,0,0] => 2 [1,1,0,1,1,0,0,0,1,0] => 5 [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] => 7 [1,1,1,0,0,0,1,1,0,0] => 3 [1,1,1,0,0,1,0,0,1,0] => 6 [1,1,1,0,0,1,0,1,0,0] => 3 [1,1,1,0,0,1,1,0,0,0] => 2 [1,1,1,0,1,0,0,0,1,0] => 5 [1,1,1,0,1,0,0,1,0,0] => 3 [1,1,1,0,1,0,1,0,0,0] => 2 [1,1,1,0,1,1,0,0,0,0] => 1 [1,1,1,1,0,0,0,0,1,0] => 4 [1,1,1,1,0,0,0,1,0,0] => 3 [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] => 10 [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] => 7 [1,0,1,0,1,1,0,1,0,0,1,0] => 11 [1,0,1,0,1,1,0,1,0,1,0,0] => 7 [1,0,1,0,1,1,0,1,1,0,0,0] => 6 [1,0,1,0,1,1,1,0,0,0,1,0] => 8 [1,0,1,0,1,1,1,0,0,1,0,0] => 7 [1,0,1,0,1,1,1,0,1,0,0,0] => 6 [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] => 8 [1,0,1,1,0,0,1,1,0,0,1,0] => 9 [1,0,1,1,0,0,1,1,0,1,0,0] => 8 [1,0,1,1,0,0,1,1,1,0,0,0] => 4 [1,0,1,1,0,1,0,0,1,0,1,0] => 12 [1,0,1,1,0,1,0,0,1,1,0,0] => 7 [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] => 8 [1,0,1,1,0,1,1,0,0,1,0,0] => 7 [1,0,1,1,0,1,1,0,1,0,0,0] => 4 [1,0,1,1,0,1,1,1,0,0,0,0] => 3 [1,0,1,1,1,0,0,0,1,0,1,0] => 10 [1,0,1,1,1,0,0,0,1,1,0,0] => 5 [1,0,1,1,1,0,0,1,0,0,1,0] => 9 [1,0,1,1,1,0,0,1,0,1,0,0] => 5 [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] => 8 [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] => 3 [1,0,1,1,1,1,0,0,0,0,1,0] => 6 [1,0,1,1,1,1,0,0,0,1,0,0] => 5 [1,0,1,1,1,1,0,0,1,0,0,0] => 4 [1,0,1,1,1,1,0,1,0,0,0,0] => 3 [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] => 9 [1,1,0,0,1,0,1,1,0,0,1,0] => 10 [1,1,0,0,1,0,1,1,0,1,0,0] => 9 [1,1,0,0,1,0,1,1,1,0,0,0] => 5 [1,1,0,0,1,1,0,0,1,0,1,0] => 11 [1,1,0,0,1,1,0,0,1,1,0,0] => 6 [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] => 6 [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] => 5 [1,1,0,0,1,1,1,1,0,0,0,0] => 2 [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] => 8 [1,1,0,1,0,0,1,1,0,0,1,0] => 9 [1,1,0,1,0,0,1,1,0,1,0,0] => 8 [1,1,0,1,0,0,1,1,1,0,0,0] => 4 [1,1,0,1,0,1,0,0,1,0,1,0] => 11 [1,1,0,1,0,1,0,0,1,1,0,0] => 6 [1,1,0,1,0,1,0,1,0,0,1,0] => 9 [1,1,0,1,0,1,0,1,0,1,0,0] => 6 [1,1,0,1,0,1,0,1,1,0,0,0] => 4 [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] => 6 [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] => 2 [1,1,0,1,1,0,0,0,1,0,1,0] => 10 [1,1,0,1,1,0,0,0,1,1,0,0] => 5 [1,1,0,1,1,0,0,1,0,0,1,0] => 9 [1,1,0,1,1,0,0,1,0,1,0,0] => 5 [1,1,0,1,1,0,0,1,1,0,0,0] => 4 [1,1,0,1,1,0,1,0,0,0,1,0] => 7 [1,1,0,1,1,0,1,0,0,1,0,0] => 5 [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] => 6 [1,1,0,1,1,1,0,0,0,1,0,0] => 5 [1,1,0,1,1,1,0,0,1,0,0,0] => 4 [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] => 12 [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] => 3 [1,1,1,0,0,1,0,0,1,0,1,0] => 11 [1,1,1,0,0,1,0,0,1,1,0,0] => 6 [1,1,1,0,0,1,0,1,0,0,1,0] => 8 [1,1,1,0,0,1,0,1,0,1,0,0] => 6 [1,1,1,0,0,1,0,1,1,0,0,0] => 3 [1,1,1,0,0,1,1,0,0,0,1,0] => 7 [1,1,1,0,0,1,1,0,0,1,0,0] => 6 [1,1,1,0,0,1,1,0,1,0,0,0] => 3 [1,1,1,0,0,1,1,1,0,0,0,0] => 2 [1,1,1,0,1,0,0,0,1,0,1,0] => 10 [1,1,1,0,1,0,0,0,1,1,0,0] => 5 [1,1,1,0,1,0,0,1,0,0,1,0] => 8 [1,1,1,0,1,0,0,1,0,1,0,0] => 5 [1,1,1,0,1,0,0,1,1,0,0,0] => 3 [1,1,1,0,1,0,1,0,0,0,1,0] => 7 [1,1,1,0,1,0,1,0,0,1,0,0] => 5 [1,1,1,0,1,0,1,0,1,0,0,0] => 3 [1,1,1,0,1,0,1,1,0,0,0,0] => 2 [1,1,1,0,1,1,0,0,0,0,1,0] => 6 [1,1,1,0,1,1,0,0,0,1,0,0] => 5 [1,1,1,0,1,1,0,0,1,0,0,0] => 3 [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] => 9 [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] => 8 [1,1,1,1,0,0,0,1,0,1,0,0] => 4 [1,1,1,1,0,0,0,1,1,0,0,0] => 3 [1,1,1,1,0,0,1,0,0,0,1,0] => 7 [1,1,1,1,0,0,1,0,0,1,0,0] => 4 [1,1,1,1,0,0,1,0,1,0,0,0] => 3 [1,1,1,1,0,0,1,1,0,0,0,0] => 2 [1,1,1,1,0,1,0,0,0,0,1,0] => 6 [1,1,1,1,0,1,0,0,0,1,0,0] => 4 [1,1,1,1,0,1,0,0,1,0,0,0] => 3 [1,1,1,1,0,1,0,1,0,0,0,0] => 2 [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] => 5 [1,1,1,1,1,0,0,0,0,1,0,0] => 4 [1,1,1,1,1,0,0,0,1,0,0,0] => 3 [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 15, 2011 at 15:56 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Jun 17, 2019 at 17:01 by Christian Stump