***************************************************************************** * 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: St001263 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The index of the maximal parabolic seaweed algebra associated with the composition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows: * place $n$ dots on a horizontal line * subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is $$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$ where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander. This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$. ----------------------------------------------------------------------------- References: [1] Dergachev, V., Kirillov, A. Index of Lie algebras of seaweed type [[MathSciNet:1774864]] [2] Coll, V., Mayers, A., Mayers, N. Statistics on integer partitions arising from seaweed algebras [[arXiv:1809.09271]] ----------------------------------------------------------------------------- Code: def to_meander(la, mu): """ Return the meander associated with two partitions as a graph. """ la = Composition(la) mu = Composition(mu) n = la.size() assert mu.size() == n M = Graph(n) offset = 0 for part in la: M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)]) offset += part offset = 0 for part in mu: M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)]) offset += part return M def seaweed_index(la, mu): M = to_meander(la, mu) index = -1 for G in M.connected_components_subgraphs(): if G.is_cycle(): index += 2 else: index += 1 return index def statistic(la): return seaweed_index(la, la.size()) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,1] => 0 [2] => 0 [1,1,1] => 1 [1,2] => 0 [2,1] => 0 [3] => 1 [1,1,1,1] => 1 [1,1,2] => 0 [1,2,1] => 1 [1,3] => 0 [2,1,1] => 0 [2,2] => 1 [3,1] => 0 [4] => 1 [1,1,1,1,1] => 2 [1,1,1,2] => 1 [1,1,2,1] => 1 [1,1,3] => 1 [1,2,1,1] => 1 [1,2,2] => 0 [1,3,1] => 2 [1,4] => 0 [2,1,1,1] => 1 [2,1,2] => 2 [2,2,1] => 0 [2,3] => 0 [3,1,1] => 1 [3,2] => 0 [4,1] => 0 [5] => 2 [1,1,1,1,1,1] => 2 [1,1,1,1,2] => 1 [1,1,1,2,1] => 1 [1,1,1,3] => 1 [1,1,2,1,1] => 2 [1,1,2,2] => 1 [1,1,3,1] => 1 [1,1,4] => 0 [1,2,1,1,1] => 1 [1,2,1,2] => 0 [1,2,2,1] => 2 [1,2,3] => 0 [1,3,1,1] => 1 [1,3,2] => 0 [1,4,1] => 2 [1,5] => 0 [2,1,1,1,1] => 1 [2,1,1,2] => 2 [2,1,2,1] => 0 [2,1,3] => 0 [2,2,1,1] => 1 [2,2,2] => 2 [2,3,1] => 0 [2,4] => 1 [3,1,1,1] => 1 [3,1,2] => 0 [3,2,1] => 0 [3,3] => 2 [4,1,1] => 0 [4,2] => 1 [5,1] => 0 [6] => 2 [1,1,1,1,1,1,1] => 3 [1,1,1,1,1,2] => 2 [1,1,1,1,2,1] => 2 [1,1,1,1,3] => 2 [1,1,1,2,1,1] => 2 [1,1,1,2,2] => 1 [1,1,1,3,1] => 2 [1,1,1,4] => 1 [1,1,2,1,1,1] => 2 [1,1,2,1,2] => 1 [1,1,2,2,1] => 1 [1,1,2,3] => 1 [1,1,3,1,1] => 3 [1,1,3,2] => 2 [1,1,4,1] => 1 [1,1,5] => 1 [1,2,1,1,1,1] => 2 [1,2,1,1,2] => 1 [1,2,1,2,1] => 3 [1,2,1,3] => 1 [1,2,2,1,1] => 1 [1,2,2,2] => 0 [1,2,3,1] => 1 [1,2,4] => 2 [1,3,1,1,1] => 2 [1,3,1,2] => 1 [1,3,2,1] => 1 [1,3,3] => 1 [1,4,1,1] => 1 [1,4,2] => 0 [1,5,1] => 3 [1,6] => 0 [2,1,1,1,1,1] => 2 [2,1,1,1,2] => 3 [2,1,1,2,1] => 1 [2,1,1,3] => 1 [2,1,2,1,1] => 1 [2,1,2,2] => 2 [2,1,3,1] => 1 [2,1,4] => 0 [2,2,1,1,1] => 1 [2,2,1,2] => 2 [2,2,2,1] => 0 [2,2,3] => 0 [2,3,1,1] => 2 [2,3,2] => 3 [2,4,1] => 0 [2,5] => 0 [3,1,1,1,1] => 2 [3,1,1,2] => 1 [3,1,2,1] => 1 [3,1,3] => 3 [3,2,1,1] => 1 [3,2,2] => 0 [3,3,1] => 1 [3,4] => 0 [4,1,1,1] => 1 [4,1,2] => 0 [4,2,1] => 2 [4,3] => 0 [5,1,1] => 1 [5,2] => 0 [6,1] => 0 [7] => 3 [1,1,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,2] => 2 [1,1,1,1,1,2,1] => 2 [1,1,1,1,1,3] => 2 [1,1,1,1,2,1,1] => 2 [1,1,1,1,2,2] => 1 [1,1,1,1,3,1] => 2 [1,1,1,1,4] => 1 [1,1,1,2,1,1,1] => 3 [1,1,1,2,1,2] => 2 [1,1,1,2,2,1] => 2 [1,1,1,2,3] => 2 [1,1,1,3,1,1] => 2 [1,1,1,3,2] => 1 [1,1,1,4,1] => 1 [1,1,1,5] => 1 [1,1,2,1,1,1,1] => 2 [1,1,2,1,1,2] => 1 [1,1,2,1,2,1] => 1 [1,1,2,1,3] => 1 [1,1,2,2,1,1] => 3 [1,1,2,2,2] => 2 [1,1,2,3,1] => 1 [1,1,2,4] => 0 [1,1,3,1,1,1] => 2 [1,1,3,1,2] => 1 [1,1,3,2,1] => 1 [1,1,3,3] => 1 [1,1,4,1,1] => 3 [1,1,4,2] => 2 [1,1,5,1] => 1 [1,1,6] => 0 [1,2,1,1,1,1,1] => 2 [1,2,1,1,1,2] => 1 [1,2,1,1,2,1] => 3 [1,2,1,1,3] => 1 [1,2,1,2,1,1] => 1 [1,2,1,2,2] => 0 [1,2,1,3,1] => 1 [1,2,1,4] => 2 [1,2,2,1,1,1] => 2 [1,2,2,1,2] => 1 [1,2,2,2,1] => 3 [1,2,2,3] => 1 [1,2,3,1,1] => 1 [1,2,3,2] => 0 [1,2,4,1] => 2 [1,2,5] => 0 [1,3,1,1,1,1] => 2 [1,3,1,1,2] => 1 [1,3,1,2,1] => 1 [1,3,1,3] => 1 [1,3,2,1,1] => 1 [1,3,2,2] => 0 [1,3,3,1] => 3 [1,3,4] => 0 [1,4,1,1,1] => 1 [1,4,1,2] => 0 [1,4,2,1] => 2 [1,4,3] => 0 [1,5,1,1] => 1 [1,5,2] => 0 [1,6,1] => 3 [1,7] => 0 [2,1,1,1,1,1,1] => 2 [2,1,1,1,1,2] => 3 [2,1,1,1,2,1] => 1 [2,1,1,1,3] => 1 [2,1,1,2,1,1] => 1 [2,1,1,2,2] => 2 [2,1,1,3,1] => 1 [2,1,1,4] => 0 [2,1,2,1,1,1] => 2 [2,1,2,1,2] => 3 [2,1,2,2,1] => 1 [2,1,2,3] => 1 [2,1,3,1,1] => 1 [2,1,3,2] => 2 [2,1,4,1] => 0 [2,1,5] => 2 [2,2,1,1,1,1] => 1 [2,2,1,1,2] => 2 [2,2,1,2,1] => 0 [2,2,1,3] => 0 [2,2,2,1,1] => 2 [2,2,2,2] => 3 [2,2,3,1] => 0 [2,2,4] => 1 [2,3,1,1,1] => 1 [2,3,1,2] => 2 [2,3,2,1] => 0 [2,3,3] => 0 [2,4,1,1] => 2 [2,4,2] => 3 [2,5,1] => 0 [2,6] => 1 [3,1,1,1,1,1] => 2 [3,1,1,1,2] => 1 [3,1,1,2,1] => 1 [3,1,1,3] => 3 [3,1,2,1,1] => 1 [3,1,2,2] => 0 [3,1,3,1] => 1 [3,1,4] => 0 [3,2,1,1,1] => 2 [3,2,1,2] => 1 [3,2,2,1] => 1 [3,2,3] => 3 [3,3,1,1] => 1 [3,3,2] => 0 [3,4,1] => 0 [3,5] => 0 [4,1,1,1,1] => 1 [4,1,1,2] => 0 [4,1,2,1] => 2 [4,1,3] => 0 [4,2,1,1] => 0 [4,2,2] => 1 [4,3,1] => 0 [4,4] => 3 [5,1,1,1] => 1 [5,1,2] => 2 [5,2,1] => 0 [5,3] => 0 [6,1,1] => 0 [6,2] => 1 [7,1] => 0 [8] => 3 [1,1,1,1,1,1,1,1,1] => 4 [1,1,1,1,1,1,1,2] => 3 [1,1,1,1,1,1,2,1] => 3 [1,1,1,1,1,1,3] => 3 [1,1,1,1,1,2,1,1] => 3 [1,1,1,1,1,2,2] => 2 [1,1,1,1,1,3,1] => 3 [1,1,1,1,1,4] => 2 [1,1,1,1,2,1,1,1] => 3 [1,1,1,1,2,1,2] => 2 [1,1,1,1,2,2,1] => 2 [1,1,1,1,2,3] => 2 [1,1,1,1,3,1,1] => 3 [1,1,1,1,3,2] => 2 [1,1,1,1,4,1] => 2 [1,1,1,1,5] => 2 [1,1,1,2,1,1,1,1] => 3 [1,1,1,2,1,1,2] => 2 [1,1,1,2,1,2,1] => 2 [1,1,1,2,1,3] => 2 [1,1,1,2,2,1,1] => 2 [1,1,1,2,2,2] => 1 [1,1,1,2,3,1] => 2 [1,1,1,2,4] => 1 [1,1,1,3,1,1,1] => 4 [1,1,1,3,1,2] => 3 [1,1,1,3,2,1] => 3 [1,1,1,3,3] => 3 [1,1,1,4,1,1] => 2 [1,1,1,4,2] => 1 [1,1,1,5,1] => 2 [1,1,1,6] => 1 [1,1,2,1,1,1,1,1] => 3 [1,1,2,1,1,1,2] => 2 [1,1,2,1,1,2,1] => 2 [1,1,2,1,1,3] => 2 [1,1,2,1,2,1,1] => 4 [1,1,2,1,2,2] => 3 [1,1,2,1,3,1] => 2 [1,1,2,1,4] => 1 [1,1,2,2,1,1,1] => 2 [1,1,2,2,1,2] => 1 [1,1,2,2,2,1] => 1 [1,1,2,2,3] => 1 [1,1,2,3,1,1] => 2 [1,1,2,3,2] => 1 [1,1,2,4,1] => 3 [1,1,2,5] => 1 [1,1,3,1,1,1,1] => 3 [1,1,3,1,1,2] => 2 [1,1,3,1,2,1] => 2 [1,1,3,1,3] => 2 [1,1,3,2,1,1] => 2 [1,1,3,2,2] => 1 [1,1,3,3,1] => 2 [1,1,3,4] => 1 [1,1,4,1,1,1] => 2 [1,1,4,1,2] => 1 [1,1,4,2,1] => 1 [1,1,4,3] => 1 [1,1,5,1,1] => 4 [1,1,5,2] => 3 [1,1,6,1] => 1 [1,1,7] => 1 [1,2,1,1,1,1,1,1] => 3 [1,2,1,1,1,1,2] => 2 [1,2,1,1,1,2,1] => 4 [1,2,1,1,1,3] => 2 [1,2,1,1,2,1,1] => 2 [1,2,1,1,2,2] => 1 [1,2,1,1,3,1] => 2 [1,2,1,1,4] => 3 [1,2,1,2,1,1,1] => 2 [1,2,1,2,1,2] => 1 [1,2,1,2,2,1] => 3 [1,2,1,2,3] => 1 [1,2,1,3,1,1] => 2 [1,2,1,3,2] => 1 [1,2,1,4,1] => 1 [1,2,1,5] => 1 [1,2,2,1,1,1,1] => 2 [1,2,2,1,1,2] => 1 [1,2,2,1,2,1] => 3 [1,2,2,1,3] => 1 [1,2,2,2,1,1] => 1 [1,2,2,2,2] => 0 [1,2,2,3,1] => 1 [1,2,2,4] => 2 [1,2,3,1,1,1] => 3 [1,2,3,1,2] => 2 [1,2,3,2,1] => 4 [1,2,3,3] => 2 [1,2,4,1,1] => 1 [1,2,4,2] => 0 [1,2,5,1] => 1 [1,2,6] => 0 [1,3,1,1,1,1,1] => 3 [1,3,1,1,1,2] => 2 [1,3,1,1,2,1] => 2 [1,3,1,1,3] => 2 [1,3,1,2,1,1] => 2 [1,3,1,2,2] => 1 [1,3,1,3,1] => 4 [1,3,1,4] => 1 [1,3,2,1,1,1] => 2 [1,3,2,1,2] => 1 [1,3,2,2,1] => 1 [1,3,2,3] => 1 [1,3,3,1,1] => 2 [1,3,3,2] => 1 [1,3,4,1] => 1 [1,3,5] => 3 [1,4,1,1,1,1] => 2 [1,4,1,1,2] => 1 [1,4,1,2,1] => 1 [1,4,1,3] => 1 [1,4,2,1,1] => 3 [1,4,2,2] => 2 [1,4,3,1] => 1 [1,4,4] => 0 [1,5,1,1,1] => 2 [1,5,1,2] => 1 [1,5,2,1] => 1 [1,5,3] => 1 [1,6,1,1] => 1 [1,6,2] => 0 [1,7,1] => 4 [1,8] => 0 [2,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,2] => 4 [2,1,1,1,1,2,1] => 2 [2,1,1,1,1,3] => 2 [2,1,1,1,2,1,1] => 2 [2,1,1,1,2,2] => 3 [2,1,1,1,3,1] => 2 [2,1,1,1,4] => 1 [2,1,1,2,1,1,1] => 2 [2,1,1,2,1,2] => 3 [2,1,1,2,2,1] => 1 [2,1,1,2,3] => 1 [2,1,1,3,1,1] => 2 [2,1,1,3,2] => 3 [2,1,1,4,1] => 1 [2,1,1,5] => 1 [2,1,2,1,1,1,1] => 2 [2,1,2,1,1,2] => 3 [2,1,2,1,2,1] => 1 [2,1,2,1,3] => 1 [2,1,2,2,1,1] => 1 [2,1,2,2,2] => 2 [2,1,2,3,1] => 1 [2,1,2,4] => 0 [2,1,3,1,1,1] => 3 [2,1,3,1,2] => 4 [2,1,3,2,1] => 2 [2,1,3,3] => 2 [2,1,4,1,1] => 1 [2,1,4,2] => 2 [2,1,5,1] => 1 [2,1,6] => 0 [2,2,1,1,1,1,1] => 2 [2,2,1,1,1,2] => 3 [2,2,1,1,2,1] => 1 [2,2,1,1,3] => 1 [2,2,1,2,1,1] => 3 [2,2,1,2,2] => 4 [2,2,1,3,1] => 1 [2,2,1,4] => 2 [2,2,2,1,1,1] => 1 [2,2,2,1,2] => 2 [2,2,2,2,1] => 0 [2,2,2,3] => 0 [2,2,3,1,1] => 1 [2,2,3,2] => 2 [2,2,4,1] => 2 [2,2,5] => 0 [2,3,1,1,1,1] => 2 [2,3,1,1,2] => 3 [2,3,1,2,1] => 1 [2,3,1,3] => 1 [2,3,2,1,1] => 1 [2,3,2,2] => 2 [2,3,3,1] => 1 [2,3,4] => 0 [2,4,1,1,1] => 1 [2,4,1,2] => 2 [2,4,2,1] => 0 [2,4,3] => 0 [2,5,1,1] => 3 [2,5,2] => 4 [2,6,1] => 0 [2,7] => 0 [3,1,1,1,1,1,1] => 3 [3,1,1,1,1,2] => 2 [3,1,1,1,2,1] => 2 [3,1,1,1,3] => 4 [3,1,1,2,1,1] => 2 [3,1,1,2,2] => 1 [3,1,1,3,1] => 2 [3,1,1,4] => 1 [3,1,2,1,1,1] => 2 [3,1,2,1,2] => 1 [3,1,2,2,1] => 1 [3,1,2,3] => 3 [3,1,3,1,1] => 2 [3,1,3,2] => 1 [3,1,4,1] => 1 [3,1,5] => 1 [3,2,1,1,1,1] => 2 [3,2,1,1,2] => 1 [3,2,1,2,1] => 1 [3,2,1,3] => 3 [3,2,2,1,1] => 1 [3,2,2,2] => 0 [3,2,3,1] => 1 [3,2,4] => 0 [3,3,1,1,1] => 3 [3,3,1,2] => 2 [3,3,2,1] => 2 [3,3,3] => 4 [3,4,1,1] => 1 [3,4,2] => 0 [3,5,1] => 1 [3,6] => 2 [4,1,1,1,1,1] => 2 [4,1,1,1,2] => 1 [4,1,1,2,1] => 3 [4,1,1,3] => 1 [4,1,2,1,1] => 1 [4,1,2,2] => 2 [4,1,3,1] => 1 [4,1,4] => 4 [4,2,1,1,1] => 1 [4,2,1,2] => 0 [4,2,2,1] => 2 [4,2,3] => 0 [4,3,1,1] => 1 [4,3,2] => 0 [4,4,1] => 0 [4,5] => 0 [5,1,1,1,1] => 2 [5,1,1,2] => 1 [5,1,2,1] => 1 [5,1,3] => 1 [5,2,1,1] => 1 [5,2,2] => 0 [5,3,1] => 3 [5,4] => 0 [6,1,1,1] => 1 [6,1,2] => 0 [6,2,1] => 0 [6,3] => 2 [7,1,1] => 1 [7,2] => 0 [8,1] => 0 [9] => 4 [1,1,1,1,1,1,1,1,1,1] => 4 [1,1,1,1,1,1,1,1,2] => 3 [1,1,1,1,1,1,1,2,1] => 3 [1,1,1,1,1,1,1,3] => 3 [1,1,1,1,1,1,2,1,1] => 3 [1,1,1,1,1,1,2,2] => 2 [1,1,1,1,1,1,3,1] => 3 [1,1,1,1,1,1,4] => 2 [1,1,1,1,1,2,1,1,1] => 3 [1,1,1,1,1,2,1,2] => 2 [1,1,1,1,1,2,2,1] => 2 [1,1,1,1,1,2,3] => 2 [1,1,1,1,1,3,1,1] => 3 [1,1,1,1,1,3,2] => 2 [1,1,1,1,1,4,1] => 2 [1,1,1,1,1,5] => 2 [1,1,1,1,2,1,1,1,1] => 4 [1,1,1,1,2,1,1,2] => 3 [1,1,1,1,2,1,2,1] => 3 [1,1,1,1,2,1,3] => 3 [1,1,1,1,2,2,1,1] => 3 [1,1,1,1,2,2,2] => 2 [1,1,1,1,2,3,1] => 3 [1,1,1,1,2,4] => 2 [1,1,1,1,3,1,1,1] => 3 [1,1,1,1,3,1,2] => 2 [1,1,1,1,3,2,1] => 2 [1,1,1,1,3,3] => 2 [1,1,1,1,4,1,1] => 2 [1,1,1,1,4,2] => 1 [1,1,1,1,5,1] => 2 [1,1,1,1,6] => 1 [1,1,1,2,1,1,1,1,1] => 3 [1,1,1,2,1,1,1,2] => 2 [1,1,1,2,1,1,2,1] => 2 [1,1,1,2,1,1,3] => 2 [1,1,1,2,1,2,1,1] => 2 [1,1,1,2,1,2,2] => 1 [1,1,1,2,1,3,1] => 2 [1,1,1,2,1,4] => 1 [1,1,1,2,2,1,1,1] => 4 [1,1,1,2,2,1,2] => 3 [1,1,1,2,2,2,1] => 3 [1,1,1,2,2,3] => 3 [1,1,1,2,3,1,1] => 2 [1,1,1,2,3,2] => 1 [1,1,1,2,4,1] => 1 [1,1,1,2,5] => 1 [1,1,1,3,1,1,1,1] => 3 [1,1,1,3,1,1,2] => 2 [1,1,1,3,1,2,1] => 2 [1,1,1,3,1,3] => 2 [1,1,1,3,2,1,1] => 2 [1,1,1,3,2,2] => 1 [1,1,1,3,3,1] => 2 [1,1,1,3,4] => 1 [1,1,1,4,1,1,1] => 4 [1,1,1,4,1,2] => 3 [1,1,1,4,2,1] => 3 [1,1,1,4,3] => 3 [1,1,1,5,1,1] => 2 [1,1,1,5,2] => 1 [1,1,1,6,1] => 1 [1,1,1,7] => 1 [1,1,2,1,1,1,1,1,1] => 3 [1,1,2,1,1,1,1,2] => 2 [1,1,2,1,1,1,2,1] => 2 [1,1,2,1,1,1,3] => 2 [1,1,2,1,1,2,1,1] => 4 [1,1,2,1,1,2,2] => 3 [1,1,2,1,1,3,1] => 2 [1,1,2,1,1,4] => 1 [1,1,2,1,2,1,1,1] => 2 [1,1,2,1,2,1,2] => 1 [1,1,2,1,2,2,1] => 1 [1,1,2,1,2,3] => 1 [1,1,2,1,3,1,1] => 2 [1,1,2,1,3,2] => 1 [1,1,2,1,4,1] => 3 [1,1,2,1,5] => 1 [1,1,2,2,1,1,1,1] => 3 [1,1,2,2,1,1,2] => 2 [1,1,2,2,1,2,1] => 2 [1,1,2,2,1,3] => 2 [1,1,2,2,2,1,1] => 4 [1,1,2,2,2,2] => 3 [1,1,2,2,3,1] => 2 [1,1,2,2,4] => 1 [1,1,2,3,1,1,1] => 2 [1,1,2,3,1,2] => 1 [1,1,2,3,2,1] => 1 [1,1,2,3,3] => 1 [1,1,2,4,1,1] => 3 [1,1,2,4,2] => 2 [1,1,2,5,1] => 1 [1,1,2,6] => 2 [1,1,3,1,1,1,1,1] => 3 [1,1,3,1,1,1,2] => 2 [1,1,3,1,1,2,1] => 2 [1,1,3,1,1,3] => 2 [1,1,3,1,2,1,1] => 2 [1,1,3,1,2,2] => 1 [1,1,3,1,3,1] => 2 [1,1,3,1,4] => 1 [1,1,3,2,1,1,1] => 2 [1,1,3,2,1,2] => 1 [1,1,3,2,2,1] => 1 [1,1,3,2,3] => 1 [1,1,3,3,1,1] => 4 [1,1,3,3,2] => 3 [1,1,3,4,1] => 1 [1,1,3,5] => 1 [1,1,4,1,1,1,1] => 2 [1,1,4,1,1,2] => 1 [1,1,4,1,2,1] => 1 [1,1,4,1,3] => 1 [1,1,4,2,1,1] => 3 [1,1,4,2,2] => 2 [1,1,4,3,1] => 1 [1,1,4,4] => 0 [1,1,5,1,1,1] => 2 [1,1,5,1,2] => 1 [1,1,5,2,1] => 1 [1,1,5,3] => 1 [1,1,6,1,1] => 4 [1,1,6,2] => 3 [1,1,7,1] => 1 [1,1,8] => 0 [1,2,1,1,1,1,1,1,1] => 3 [1,2,1,1,1,1,1,2] => 2 [1,2,1,1,1,1,2,1] => 4 [1,2,1,1,1,1,3] => 2 [1,2,1,1,1,2,1,1] => 2 [1,2,1,1,1,2,2] => 1 [1,2,1,1,1,3,1] => 2 [1,2,1,1,1,4] => 3 [1,2,1,1,2,1,1,1] => 2 [1,2,1,1,2,1,2] => 1 [1,2,1,1,2,2,1] => 3 [1,2,1,1,2,3] => 1 [1,2,1,1,3,1,1] => 2 [1,2,1,1,3,2] => 1 [1,2,1,1,4,1] => 1 [1,2,1,1,5] => 1 [1,2,1,2,1,1,1,1] => 3 [1,2,1,2,1,1,2] => 2 [1,2,1,2,1,2,1] => 4 [1,2,1,2,1,3] => 2 [1,2,1,2,2,1,1] => 2 [1,2,1,2,2,2] => 1 [1,2,1,2,3,1] => 2 [1,2,1,2,4] => 3 [1,2,1,3,1,1,1] => 2 [1,2,1,3,1,2] => 1 [1,2,1,3,2,1] => 3 [1,2,1,3,3] => 1 [1,2,1,4,1,1] => 1 [1,2,1,4,2] => 0 [1,2,1,5,1] => 3 [1,2,1,6] => 0 [1,2,2,1,1,1,1,1] => 2 [1,2,2,1,1,1,2] => 1 [1,2,2,1,1,2,1] => 3 [1,2,2,1,1,3] => 1 [1,2,2,1,2,1,1] => 1 [1,2,2,1,2,2] => 0 [1,2,2,1,3,1] => 1 [1,2,2,1,4] => 2 [1,2,2,2,1,1,1] => 3 [1,2,2,2,1,2] => 2 [1,2,2,2,2,1] => 4 [1,2,2,2,3] => 2 [1,2,2,3,1,1] => 1 [1,2,2,3,2] => 0 [1,2,2,4,1] => 2 [1,2,2,5] => 0 [1,2,3,1,1,1,1] => 2 [1,2,3,1,1,2] => 1 [1,2,3,1,2,1] => 3 [1,2,3,1,3] => 1 [1,2,3,2,1,1] => 1 [1,2,3,2,2] => 0 [1,2,3,3,1] => 1 [1,2,3,4] => 2 [1,2,4,1,1,1] => 3 [1,2,4,1,2] => 2 [1,2,4,2,1] => 4 [1,2,4,3] => 2 [1,2,5,1,1] => 1 [1,2,5,2] => 0 [1,2,6,1] => 2 [1,2,7] => 2 [1,3,1,1,1,1,1,1] => 3 [1,3,1,1,1,1,2] => 2 [1,3,1,1,1,2,1] => 2 [1,3,1,1,1,3] => 2 [1,3,1,1,2,1,1] => 2 [1,3,1,1,2,2] => 1 [1,3,1,1,3,1] => 4 [1,3,1,1,4] => 1 [1,3,1,2,1,1,1] => 2 [1,3,1,2,1,2] => 1 [1,3,1,2,2,1] => 1 [1,3,1,2,3] => 1 [1,3,1,3,1,1] => 2 [1,3,1,3,2] => 1 [1,3,1,4,1] => 1 [1,3,1,5] => 3 [1,3,2,1,1,1,1] => 3 [1,3,2,1,1,2] => 2 [1,3,2,1,2,1] => 2 [1,3,2,1,3] => 2 [1,3,2,2,1,1] => 2 [1,3,2,2,2] => 1 [1,3,2,3,1] => 4 [1,3,2,4] => 1 [1,3,3,1,1,1] => 2 [1,3,3,1,2] => 1 [1,3,3,2,1] => 1 [1,3,3,3] => 1 [1,3,4,1,1] => 1 [1,3,4,2] => 0 [1,3,5,1] => 1 [1,3,6] => 0 [1,4,1,1,1,1,1] => 2 [1,4,1,1,1,2] => 1 [1,4,1,1,2,1] => 1 [1,4,1,1,3] => 1 [1,4,1,2,1,1] => 3 [1,4,1,2,2] => 2 [1,4,1,3,1] => 1 [1,4,1,4] => 0 [1,4,2,1,1,1] => 1 [1,4,2,1,2] => 0 [1,4,2,2,1] => 2 [1,4,2,3] => 0 [1,4,3,1,1] => 1 [1,4,3,2] => 0 [1,4,4,1] => 4 [1,4,5] => 0 [1,5,1,1,1,1] => 2 [1,5,1,1,2] => 1 [1,5,1,2,1] => 3 [1,5,1,3] => 1 [1,5,2,1,1] => 1 [1,5,2,2] => 0 [1,5,3,1] => 1 [1,5,4] => 2 [1,6,1,1,1] => 1 [1,6,1,2] => 0 [1,6,2,1] => 2 [1,6,3] => 0 [1,7,1,1] => 1 [1,7,2] => 0 [1,8,1] => 4 [1,9] => 0 [2,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,2] => 4 [2,1,1,1,1,1,2,1] => 2 [2,1,1,1,1,1,3] => 2 [2,1,1,1,1,2,1,1] => 2 [2,1,1,1,1,2,2] => 3 [2,1,1,1,1,3,1] => 2 [2,1,1,1,1,4] => 1 [2,1,1,1,2,1,1,1] => 2 [2,1,1,1,2,1,2] => 3 [2,1,1,1,2,2,1] => 1 [2,1,1,1,2,3] => 1 [2,1,1,1,3,1,1] => 2 [2,1,1,1,3,2] => 3 [2,1,1,1,4,1] => 1 [2,1,1,1,5] => 1 [2,1,1,2,1,1,1,1] => 3 [2,1,1,2,1,1,2] => 4 [2,1,1,2,1,2,1] => 2 [2,1,1,2,1,3] => 2 [2,1,1,2,2,1,1] => 2 [2,1,1,2,2,2] => 3 [2,1,1,2,3,1] => 2 [2,1,1,2,4] => 1 [2,1,1,3,1,1,1] => 2 [2,1,1,3,1,2] => 3 [2,1,1,3,2,1] => 1 [2,1,1,3,3] => 1 [2,1,1,4,1,1] => 1 [2,1,1,4,2] => 2 [2,1,1,5,1] => 1 [2,1,1,6] => 2 [2,1,2,1,1,1,1,1] => 2 [2,1,2,1,1,1,2] => 3 [2,1,2,1,1,2,1] => 1 [2,1,2,1,1,3] => 1 [2,1,2,1,2,1,1] => 1 [2,1,2,1,2,2] => 2 [2,1,2,1,3,1] => 1 [2,1,2,1,4] => 0 [2,1,2,2,1,1,1] => 3 [2,1,2,2,1,2] => 4 [2,1,2,2,2,1] => 2 [2,1,2,2,3] => 2 [2,1,2,3,1,1] => 1 [2,1,2,3,2] => 2 [2,1,2,4,1] => 0 [2,1,2,5] => 2 [2,1,3,1,1,1,1] => 2 [2,1,3,1,1,2] => 3 [2,1,3,1,2,1] => 1 [2,1,3,1,3] => 1 [2,1,3,2,1,1] => 1 [2,1,3,2,2] => 2 [2,1,3,3,1] => 1 [2,1,3,4] => 0 [2,1,4,1,1,1] => 3 [2,1,4,1,2] => 4 [2,1,4,2,1] => 2 [2,1,4,3] => 2 [2,1,5,1,1] => 1 [2,1,5,2] => 2 [2,1,6,1] => 0 [2,1,7] => 0 [2,2,1,1,1,1,1,1] => 2 [2,2,1,1,1,1,2] => 3 [2,2,1,1,1,2,1] => 1 [2,2,1,1,1,3] => 1 [2,2,1,1,2,1,1] => 3 [2,2,1,1,2,2] => 4 [2,2,1,1,3,1] => 1 [2,2,1,1,4] => 2 [2,2,1,2,1,1,1] => 1 [2,2,1,2,1,2] => 2 [2,2,1,2,2,1] => 0 [2,2,1,2,3] => 0 [2,2,1,3,1,1] => 1 [2,2,1,3,2] => 2 [2,2,1,4,1] => 2 [2,2,1,5] => 0 [2,2,2,1,1,1,1] => 2 [2,2,2,1,1,2] => 3 [2,2,2,1,2,1] => 1 [2,2,2,1,3] => 1 [2,2,2,2,1,1] => 3 [2,2,2,2,2] => 4 [2,2,2,3,1] => 1 [2,2,2,4] => 2 [2,2,3,1,1,1] => 1 [2,2,3,1,2] => 2 [2,2,3,2,1] => 0 [2,2,3,3] => 0 [2,2,4,1,1] => 2 [2,2,4,2] => 3 [2,2,5,1] => 0 [2,2,6] => 3 [2,3,1,1,1,1,1] => 2 [2,3,1,1,1,2] => 3 [2,3,1,1,2,1] => 1 [2,3,1,1,3] => 1 [2,3,1,2,1,1] => 1 [2,3,1,2,2] => 2 [2,3,1,3,1] => 1 [2,3,1,4] => 0 [2,3,2,1,1,1] => 1 [2,3,2,1,2] => 2 [2,3,2,2,1] => 0 [2,3,2,3] => 0 [2,3,3,1,1] => 3 [2,3,3,2] => 4 [2,3,4,1] => 0 [2,3,5] => 0 [2,4,1,1,1,1] => 1 [2,4,1,1,2] => 2 [2,4,1,2,1] => 0 [2,4,1,3] => 0 [2,4,2,1,1] => 2 [2,4,2,2] => 3 [2,4,3,1] => 0 [2,4,4] => 1 [2,5,1,1,1] => 1 [2,5,1,2] => 2 [2,5,2,1] => 0 [2,5,3] => 0 [2,6,1,1] => 3 [2,6,2] => 4 [2,7,1] => 0 [2,8] => 1 [3,1,1,1,1,1,1,1] => 3 [3,1,1,1,1,1,2] => 2 [3,1,1,1,1,2,1] => 2 [3,1,1,1,1,3] => 4 [3,1,1,1,2,1,1] => 2 [3,1,1,1,2,2] => 1 [3,1,1,1,3,1] => 2 [3,1,1,1,4] => 1 [3,1,1,2,1,1,1] => 2 [3,1,1,2,1,2] => 1 [3,1,1,2,2,1] => 1 [3,1,1,2,3] => 3 [3,1,1,3,1,1] => 2 [3,1,1,3,2] => 1 [3,1,1,4,1] => 1 [3,1,1,5] => 1 [3,1,2,1,1,1,1] => 3 [3,1,2,1,1,2] => 2 [3,1,2,1,2,1] => 2 [3,1,2,1,3] => 4 [3,1,2,2,1,1] => 2 [3,1,2,2,2] => 1 [3,1,2,3,1] => 2 [3,1,2,4] => 1 [3,1,3,1,1,1] => 2 [3,1,3,1,2] => 1 [3,1,3,2,1] => 1 [3,1,3,3] => 3 [3,1,4,1,1] => 1 [3,1,4,2] => 0 [3,1,5,1] => 1 [3,1,6] => 0 [3,2,1,1,1,1,1] => 2 [3,2,1,1,1,2] => 1 [3,2,1,1,2,1] => 1 [3,2,1,1,3] => 3 [3,2,1,2,1,1] => 1 [3,2,1,2,2] => 0 [3,2,1,3,1] => 1 [3,2,1,4] => 0 [3,2,2,1,1,1] => 3 [3,2,2,1,2] => 2 [3,2,2,2,1] => 2 [3,2,2,3] => 4 [3,2,3,1,1] => 1 [3,2,3,2] => 0 [3,2,4,1] => 0 [3,2,5] => 0 [3,3,1,1,1,1] => 2 [3,3,1,1,2] => 1 [3,3,1,2,1] => 1 [3,3,1,3] => 3 [3,3,2,1,1] => 1 [3,3,2,2] => 0 [3,3,3,1] => 1 [3,3,4] => 0 [3,4,1,1,1] => 3 [3,4,1,2] => 2 [3,4,2,1] => 2 [3,4,3] => 4 [3,5,1,1] => 1 [3,5,2] => 0 [3,6,1] => 0 [3,7] => 0 [4,1,1,1,1,1,1] => 2 [4,1,1,1,1,2] => 1 [4,1,1,1,2,1] => 3 [4,1,1,1,3] => 1 [4,1,1,2,1,1] => 1 [4,1,1,2,2] => 2 [4,1,1,3,1] => 1 [4,1,1,4] => 4 [4,1,2,1,1,1] => 1 [4,1,2,1,2] => 0 [4,1,2,2,1] => 2 [4,1,2,3] => 0 [4,1,3,1,1] => 1 [4,1,3,2] => 0 [4,1,4,1] => 0 [4,1,5] => 0 [4,2,1,1,1,1] => 2 [4,2,1,1,2] => 1 [4,2,1,2,1] => 3 [4,2,1,3] => 1 [4,2,2,1,1] => 1 [4,2,2,2] => 2 [4,2,3,1] => 1 [4,2,4] => 4 [4,3,1,1,1] => 1 [4,3,1,2] => 0 [4,3,2,1] => 2 [4,3,3] => 0 [4,4,1,1] => 0 [4,4,2] => 1 [4,5,1] => 2 [4,6] => 1 [5,1,1,1,1,1] => 2 [5,1,1,1,2] => 1 [5,1,1,2,1] => 1 [5,1,1,3] => 1 [5,1,2,1,1] => 1 [5,1,2,2] => 0 [5,1,3,1] => 3 [5,1,4] => 0 [5,2,1,1,1] => 1 [5,2,1,2] => 2 [5,2,2,1] => 0 [5,2,3] => 0 [5,3,1,1] => 1 [5,3,2] => 0 [5,4,1] => 0 [5,5] => 4 [6,1,1,1,1] => 1 [6,1,1,2] => 2 [6,1,2,1] => 0 [6,1,3] => 0 [6,2,1,1] => 2 [6,2,2] => 3 [6,3,1] => 0 [6,4] => 1 [7,1,1,1] => 1 [7,1,2] => 0 [7,2,1] => 2 [7,3] => 0 [8,1,1] => 0 [8,2] => 1 [9,1] => 0 [10] => 4 ----------------------------------------------------------------------------- Created: Sep 26, 2018 at 04:49 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 26, 2018 at 10:46 by Martin Rubey