***************************************************************************** * 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: St000174 ----------------------------------------------------------------------------- Collection: Semistandard tableaux ----------------------------------------------------------------------------- Description: The flush statistic of a semistandard tableau. Let $T$ be a tableaux with $r$ rows such that each row is longer than the row beneath it by at least one box. Let $1 \leq i < k \leq r+1$ and suppose $l$ is the smallest integer greater than $k$ such that there exists an $l$-segment in the $(i+1)$-st row of $T$. A $k$-segment in the $i$-th row of $T$ is called '''flush''' if the leftmost box in the $k$-segment and the leftmost box of the $l$-segment are in the same column of $T$. If, however, no such $l$ exists, then this $k$-segment is said to be flush if the number of boxes in the $k$-segment is equal to difference of the number of boxes between the $i$-th row and $(i+1)$-st row. The flush statistic is given by the number of $k$-segments in $T$. ----------------------------------------------------------------------------- References: [1] Salisbury, B. The flush statistic on semistandard Young tableaux [[arXiv:1401.1185]] ----------------------------------------------------------------------------- Code: def statistic(T): f = 0 segments = {} for r in range(len(T)): for c in range(len(T[r])): for j in range(c+1): if T[r][j] != r+1: if (r,T[r][j]) not in segments.keys(): segments[(r,T[r][j])] = j L = segments.items() for s in segments.iteritems(): if s[0][0] != len(T)-1 and s[1] == len(T[s[0][0]+1]) and T[s[0][0]+1][-1] <= s[0][1]: f += 1 if s[0][0] == len(T)-1 and s[1] == 0: f += 1 else: for t in L: if s[0][0]+1 == t[0][0] and s[1] == t[1]: if s[1] >= 1 and T[s[0][0]+1][s[1]] != T[s[0][0]+1][s[1]-1]: f += 1 if s[1] < 1 and T[s[0][0]+1][s[1]] != s[0][0]+2: f += 1 return f ----------------------------------------------------------------------------- Statistic values: [[1]] => 0 [[2]] => 1 [[3]] => 1 [[4]] => 1 [[1,1]] => 0 [[1,2]] => 0 [[2,2]] => 1 [[1],[2]] => 0 [[1,3]] => 0 [[2,3]] => 1 [[3,3]] => 1 [[1],[3]] => 1 [[2],[3]] => 2 [[1,4]] => 0 [[2,4]] => 1 [[3,4]] => 1 [[4,4]] => 1 [[1],[4]] => 1 [[2],[4]] => 2 [[3],[4]] => 2 [[1,1,1]] => 0 [[1,1,2]] => 0 [[1,2,2]] => 0 [[2,2,2]] => 1 [[1,1],[2]] => 0 [[1,2],[2]] => 1 [[1,1,3]] => 0 [[1,2,3]] => 0 [[1,3,3]] => 0 [[2,2,3]] => 1 [[2,3,3]] => 1 [[3,3,3]] => 1 [[1,1],[3]] => 1 [[1,2],[3]] => 1 [[1,3],[2]] => 1 [[1,3],[3]] => 2 [[2,2],[3]] => 2 [[2,3],[3]] => 3 [[1],[2],[3]] => 0 [[1,1,4]] => 0 [[1,2,4]] => 0 [[1,3,4]] => 0 [[1,4,4]] => 0 [[2,2,4]] => 1 [[2,3,4]] => 1 [[2,4,4]] => 1 [[3,3,4]] => 1 [[3,4,4]] => 1 [[4,4,4]] => 1 [[1,1],[4]] => 1 [[1,2],[4]] => 1 [[1,4],[2]] => 1 [[1,3],[4]] => 1 [[1,4],[3]] => 2 [[1,4],[4]] => 2 [[2,2],[4]] => 2 [[2,3],[4]] => 2 [[2,4],[3]] => 3 [[2,4],[4]] => 3 [[3,3],[4]] => 2 [[3,4],[4]] => 3 [[1],[2],[4]] => 1 [[1],[3],[4]] => 2 [[2],[3],[4]] => 3 [[1,1,1,1]] => 0 [[1,1,1,2]] => 0 [[1,1,2,2]] => 0 [[1,2,2,2]] => 0 [[2,2,2,2]] => 1 [[1,1,1],[2]] => 0 [[1,1,2],[2]] => 0 [[1,2,2],[2]] => 1 [[1,1],[2,2]] => 0 [[1,1,1,3]] => 0 [[1,1,2,3]] => 0 [[1,1,3,3]] => 0 [[1,2,2,3]] => 0 [[1,2,3,3]] => 0 [[1,3,3,3]] => 0 [[2,2,2,3]] => 1 [[2,2,3,3]] => 1 [[2,3,3,3]] => 1 [[3,3,3,3]] => 1 [[1,1,1],[3]] => 1 [[1,1,2],[3]] => 1 [[1,1,3],[2]] => 0 [[1,1,3],[3]] => 1 [[1,2,2],[3]] => 1 [[1,2,3],[2]] => 1 [[1,2,3],[3]] => 1 [[1,3,3],[2]] => 1 [[1,3,3],[3]] => 2 [[2,2,2],[3]] => 2 [[2,2,3],[3]] => 2 [[2,3,3],[3]] => 3 [[1,1],[2,3]] => 0 [[1,1],[3,3]] => 1 [[1,2],[2,3]] => 1 [[1,2],[3,3]] => 1 [[2,2],[3,3]] => 2 [[1,1],[2],[3]] => 0 [[1,2],[2],[3]] => 1 [[1,3],[2],[3]] => 1 [[1,1,1,4]] => 0 [[1,1,2,4]] => 0 [[1,1,3,4]] => 0 [[1,1,4,4]] => 0 [[1,2,2,4]] => 0 [[1,2,3,4]] => 0 [[1,2,4,4]] => 0 [[1,3,3,4]] => 0 [[1,3,4,4]] => 0 [[1,4,4,4]] => 0 [[2,2,2,4]] => 1 [[2,2,3,4]] => 1 [[2,2,4,4]] => 1 [[2,3,3,4]] => 1 [[2,3,4,4]] => 1 [[2,4,4,4]] => 1 [[3,3,3,4]] => 1 [[3,3,4,4]] => 1 [[3,4,4,4]] => 1 [[4,4,4,4]] => 1 [[1,1,1],[4]] => 1 [[1,1,2],[4]] => 1 [[1,1,4],[2]] => 0 [[1,1,3],[4]] => 1 [[1,1,4],[3]] => 1 [[1,1,4],[4]] => 1 [[1,2,2],[4]] => 1 [[1,2,4],[2]] => 1 [[1,2,3],[4]] => 1 [[1,2,4],[3]] => 1 [[1,3,4],[2]] => 1 [[1,2,4],[4]] => 1 [[1,4,4],[2]] => 1 [[1,3,3],[4]] => 1 [[1,3,4],[3]] => 2 [[1,3,4],[4]] => 1 [[1,4,4],[3]] => 2 [[1,4,4],[4]] => 2 [[2,2,2],[4]] => 2 [[2,2,3],[4]] => 2 [[2,2,4],[3]] => 2 [[2,2,4],[4]] => 2 [[2,3,3],[4]] => 2 [[2,3,4],[3]] => 3 [[2,3,4],[4]] => 2 [[2,4,4],[3]] => 3 [[2,4,4],[4]] => 3 [[3,3,3],[4]] => 2 [[3,3,4],[4]] => 2 [[3,4,4],[4]] => 3 [[1,1],[2,4]] => 0 [[1,1],[3,4]] => 1 [[1,1],[4,4]] => 1 [[1,2],[2,4]] => 1 [[1,2],[3,4]] => 2 [[1,3],[2,4]] => 1 [[1,2],[4,4]] => 1 [[1,3],[3,4]] => 2 [[1,3],[4,4]] => 1 [[2,2],[3,4]] => 2 [[2,2],[4,4]] => 2 [[2,3],[3,4]] => 3 [[2,3],[4,4]] => 2 [[3,3],[4,4]] => 2 [[1,1],[2],[4]] => 1 [[1,1],[3],[4]] => 2 [[1,2],[2],[4]] => 2 [[1,2],[3],[4]] => 2 [[1,3],[2],[4]] => 2 [[1,4],[2],[3]] => 1 [[1,4],[2],[4]] => 2 [[1,3],[3],[4]] => 3 [[1,4],[3],[4]] => 3 [[2,2],[3],[4]] => 3 [[2,3],[3],[4]] => 4 [[2,4],[3],[4]] => 4 [[1],[2],[3],[4]] => 0 [[1,1,1,1,1]] => 0 [[1,1,1,1,2]] => 0 [[1,1,1,2,2]] => 0 [[1,1,2,2,2]] => 0 [[1,2,2,2,2]] => 0 [[2,2,2,2,2]] => 1 [[1,1,1,1],[2]] => 0 [[1,1,1,2],[2]] => 0 [[1,1,2,2],[2]] => 0 [[1,2,2,2],[2]] => 1 [[1,1,1],[2,2]] => 0 [[1,1,2],[2,2]] => 1 [[1,1,1,1,3]] => 0 [[1,1,1,2,3]] => 0 [[1,1,1,3,3]] => 0 [[1,1,2,2,3]] => 0 [[1,1,2,3,3]] => 0 [[1,1,3,3,3]] => 0 [[1,2,2,2,3]] => 0 [[1,2,2,3,3]] => 0 [[1,2,3,3,3]] => 0 [[1,3,3,3,3]] => 0 [[2,2,2,2,3]] => 1 [[2,2,2,3,3]] => 1 [[2,2,3,3,3]] => 1 [[2,3,3,3,3]] => 1 [[3,3,3,3,3]] => 1 [[1,1,1,1],[3]] => 1 [[1,1,1,2],[3]] => 1 [[1,1,1,3],[2]] => 0 [[1,1,1,3],[3]] => 1 [[1,1,2,2],[3]] => 1 [[1,1,2,3],[2]] => 0 [[1,1,2,3],[3]] => 1 [[1,1,3,3],[2]] => 0 [[1,1,3,3],[3]] => 1 [[1,2,2,2],[3]] => 1 [[1,2,2,3],[2]] => 1 [[1,2,2,3],[3]] => 1 [[1,2,3,3],[2]] => 1 [[1,2,3,3],[3]] => 1 [[1,3,3,3],[2]] => 1 [[1,3,3,3],[3]] => 2 [[2,2,2,2],[3]] => 2 [[2,2,2,3],[3]] => 2 [[2,2,3,3],[3]] => 2 [[2,3,3,3],[3]] => 3 [[1,1,1],[2,3]] => 0 [[1,1,1],[3,3]] => 1 [[1,1,2],[2,3]] => 0 [[1,1,3],[2,2]] => 1 [[1,1,2],[3,3]] => 1 [[1,1,3],[2,3]] => 1 [[1,1,3],[3,3]] => 2 [[1,2,2],[2,3]] => 1 [[1,2,2],[3,3]] => 1 [[1,2,3],[2,3]] => 2 [[1,2,3],[3,3]] => 2 [[2,2,2],[3,3]] => 2 [[2,2,3],[3,3]] => 3 [[1,1,1],[2],[3]] => 0 [[1,1,2],[2],[3]] => 0 [[1,1,3],[2],[3]] => 0 [[1,2,2],[2],[3]] => 1 [[1,2,3],[2],[3]] => 1 [[1,3,3],[2],[3]] => 1 [[1,1],[2,2],[3]] => 0 [[1,1],[2,3],[3]] => 1 [[1,2],[2,3],[3]] => 2 [[1,1,1,1,4]] => 0 [[1,1,1,2,4]] => 0 [[1,1,1,3,4]] => 0 [[1,1,1,4,4]] => 0 [[1,1,2,2,4]] => 0 [[1,1,2,3,4]] => 0 [[1,1,2,4,4]] => 0 [[1,1,3,3,4]] => 0 [[1,1,3,4,4]] => 0 [[1,1,4,4,4]] => 0 [[1,2,2,2,4]] => 0 [[1,2,2,3,4]] => 0 [[1,2,2,4,4]] => 0 [[1,2,3,3,4]] => 0 [[1,2,3,4,4]] => 0 [[1,2,4,4,4]] => 0 [[1,3,3,3,4]] => 0 [[1,3,3,4,4]] => 0 [[1,3,4,4,4]] => 0 [[1,4,4,4,4]] => 0 [[2,2,2,2,4]] => 1 [[2,2,2,3,4]] => 1 [[2,2,2,4,4]] => 1 [[2,2,3,3,4]] => 1 [[2,2,3,4,4]] => 1 [[2,2,4,4,4]] => 1 [[2,3,3,3,4]] => 1 [[2,3,3,4,4]] => 1 [[2,3,4,4,4]] => 1 [[2,4,4,4,4]] => 1 [[3,3,3,3,4]] => 1 [[3,3,3,4,4]] => 1 [[3,3,4,4,4]] => 1 [[3,4,4,4,4]] => 1 [[4,4,4,4,4]] => 1 [[1,1,1,1],[4]] => 1 [[1,1,1,2],[4]] => 1 [[1,1,1,4],[2]] => 0 [[1,1,1,3],[4]] => 1 [[1,1,1,4],[3]] => 1 [[1,1,1,4],[4]] => 1 [[1,1,2,2],[4]] => 1 [[1,1,2,4],[2]] => 0 [[1,1,2,3],[4]] => 1 [[1,1,2,4],[3]] => 1 [[1,1,3,4],[2]] => 0 [[1,1,2,4],[4]] => 1 [[1,1,4,4],[2]] => 0 [[1,1,3,3],[4]] => 1 [[1,1,3,4],[3]] => 1 [[1,1,3,4],[4]] => 1 [[1,1,4,4],[3]] => 1 [[1,1,4,4],[4]] => 1 [[1,2,2,2],[4]] => 1 [[1,2,2,4],[2]] => 1 [[1,2,2,3],[4]] => 1 [[1,2,2,4],[3]] => 1 [[1,2,3,4],[2]] => 1 [[1,2,2,4],[4]] => 1 [[1,2,4,4],[2]] => 1 [[1,2,3,3],[4]] => 1 [[1,2,3,4],[3]] => 1 [[1,3,3,4],[2]] => 1 [[1,2,3,4],[4]] => 1 [[1,2,4,4],[3]] => 1 [[1,3,4,4],[2]] => 1 [[1,2,4,4],[4]] => 1 [[1,4,4,4],[2]] => 1 [[1,3,3,3],[4]] => 1 [[1,3,3,4],[3]] => 2 [[1,3,3,4],[4]] => 1 [[1,3,4,4],[3]] => 2 [[1,3,4,4],[4]] => 1 [[1,4,4,4],[3]] => 2 [[1,4,4,4],[4]] => 2 [[2,2,2,2],[4]] => 2 [[2,2,2,3],[4]] => 2 [[2,2,2,4],[3]] => 2 [[2,2,2,4],[4]] => 2 [[2,2,3,3],[4]] => 2 [[2,2,3,4],[3]] => 2 [[2,2,3,4],[4]] => 2 [[2,2,4,4],[3]] => 2 [[2,2,4,4],[4]] => 2 [[2,3,3,3],[4]] => 2 [[2,3,3,4],[3]] => 3 [[2,3,3,4],[4]] => 2 [[2,3,4,4],[3]] => 3 [[2,3,4,4],[4]] => 2 [[2,4,4,4],[3]] => 3 [[2,4,4,4],[4]] => 3 [[3,3,3,3],[4]] => 2 [[3,3,3,4],[4]] => 2 [[3,3,4,4],[4]] => 2 [[3,4,4,4],[4]] => 3 [[1,1,1],[2,4]] => 0 [[1,1,1],[3,4]] => 1 [[1,1,1],[4,4]] => 1 [[1,1,2],[2,4]] => 0 [[1,1,4],[2,2]] => 1 [[1,1,2],[3,4]] => 1 [[1,1,3],[2,4]] => 0 [[1,1,4],[2,3]] => 1 [[1,1,2],[4,4]] => 1 [[1,1,4],[2,4]] => 1 [[1,1,3],[3,4]] => 1 [[1,1,4],[3,3]] => 2 [[1,1,3],[4,4]] => 1 [[1,1,4],[3,4]] => 2 [[1,1,4],[4,4]] => 2 [[1,2,2],[2,4]] => 1 [[1,2,2],[3,4]] => 2 [[1,2,3],[2,4]] => 1 [[1,2,4],[2,3]] => 2 [[1,2,2],[4,4]] => 1 [[1,2,4],[2,4]] => 2 [[1,2,3],[3,4]] => 2 [[1,2,4],[3,3]] => 2 [[1,3,3],[2,4]] => 1 [[1,2,3],[4,4]] => 1 [[1,2,4],[3,4]] => 3 [[1,3,4],[2,4]] => 2 [[1,2,4],[4,4]] => 2 [[1,3,3],[3,4]] => 2 [[1,3,3],[4,4]] => 1 [[1,3,4],[3,4]] => 3 [[1,3,4],[4,4]] => 2 [[2,2,2],[3,4]] => 2 [[2,2,2],[4,4]] => 2 [[2,2,3],[3,4]] => 2 [[2,2,4],[3,3]] => 3 [[2,2,3],[4,4]] => 2 [[2,2,4],[3,4]] => 3 [[2,2,4],[4,4]] => 3 [[2,3,3],[3,4]] => 3 [[2,3,3],[4,4]] => 2 [[2,3,4],[3,4]] => 4 [[2,3,4],[4,4]] => 3 [[3,3,3],[4,4]] => 2 [[3,3,4],[4,4]] => 3 [[1,1,1],[2],[4]] => 1 [[1,1,1],[3],[4]] => 2 [[1,1,2],[2],[4]] => 1 [[1,1,2],[3],[4]] => 2 [[1,1,3],[2],[4]] => 1 [[1,1,4],[2],[3]] => 0 [[1,1,4],[2],[4]] => 1 [[1,1,3],[3],[4]] => 2 [[1,1,4],[3],[4]] => 2 [[1,2,2],[2],[4]] => 2 [[1,2,2],[3],[4]] => 2 [[1,2,3],[2],[4]] => 2 [[1,2,4],[2],[3]] => 1 [[1,2,4],[2],[4]] => 2 [[1,2,3],[3],[4]] => 2 [[1,3,3],[2],[4]] => 2 [[1,3,4],[2],[3]] => 1 [[1,2,4],[3],[4]] => 2 [[1,3,4],[2],[4]] => 2 [[1,4,4],[2],[3]] => 1 [[1,4,4],[2],[4]] => 2 [[1,3,3],[3],[4]] => 3 [[1,3,4],[3],[4]] => 3 [[1,4,4],[3],[4]] => 3 [[2,2,2],[3],[4]] => 3 [[2,2,3],[3],[4]] => 3 [[2,2,4],[3],[4]] => 3 [[2,3,3],[3],[4]] => 4 [[2,3,4],[3],[4]] => 4 [[2,4,4],[3],[4]] => 4 [[1,1],[2,2],[4]] => 1 [[1,1],[2,3],[4]] => 1 [[1,1],[2,4],[3]] => 1 [[1,1],[2,4],[4]] => 2 [[1,1],[3,3],[4]] => 2 [[1,1],[3,4],[4]] => 3 [[1,2],[2,3],[4]] => 2 [[1,2],[2,4],[3]] => 2 [[1,2],[2,4],[4]] => 3 [[1,2],[3,3],[4]] => 2 [[1,3],[2,4],[3]] => 2 [[1,2],[3,4],[4]] => 4 [[1,3],[2,4],[4]] => 3 [[1,3],[3,4],[4]] => 4 [[2,2],[3,3],[4]] => 3 [[2,2],[3,4],[4]] => 4 [[2,3],[3,4],[4]] => 5 [[1,1],[2],[3],[4]] => 0 [[1,2],[2],[3],[4]] => 1 [[1,3],[2],[3],[4]] => 1 [[1,4],[2],[3],[4]] => 1 [[1,1,1,1,1,1]] => 0 [[1,1,1,1,1,2]] => 0 [[1,1,1,1,2,2]] => 0 [[1,1,1,2,2,2]] => 0 [[1,1,2,2,2,2]] => 0 [[1,2,2,2,2,2]] => 0 [[2,2,2,2,2,2]] => 1 [[1,1,1,1,1],[2]] => 0 [[1,1,1,1,2],[2]] => 0 [[1,1,1,2,2],[2]] => 0 [[1,1,2,2,2],[2]] => 0 [[1,2,2,2,2],[2]] => 1 [[1,1,1,1],[2,2]] => 0 [[1,1,1,2],[2,2]] => 0 [[1,1,2,2],[2,2]] => 1 [[1,1,1],[2,2,2]] => 0 [[1,1,1,1,1,3]] => 0 [[1,1,1,1,2,3]] => 0 [[1,1,1,1,3,3]] => 0 [[1,1,1,2,2,3]] => 0 [[1,1,1,2,3,3]] => 0 [[1,1,1,3,3,3]] => 0 [[1,1,2,2,2,3]] => 0 [[1,1,2,2,3,3]] => 0 [[1,1,2,3,3,3]] => 0 [[1,1,3,3,3,3]] => 0 [[1,2,2,2,2,3]] => 0 [[1,2,2,2,3,3]] => 0 [[1,2,2,3,3,3]] => 0 [[1,2,3,3,3,3]] => 0 [[1,3,3,3,3,3]] => 0 [[2,2,2,2,2,3]] => 1 [[2,2,2,2,3,3]] => 1 [[2,2,2,3,3,3]] => 1 [[2,2,3,3,3,3]] => 1 [[2,3,3,3,3,3]] => 1 [[3,3,3,3,3,3]] => 1 [[1,1,1,1,1],[3]] => 1 [[1,1,1,1,2],[3]] => 1 [[1,1,1,1,3],[2]] => 0 [[1,1,1,1,3],[3]] => 1 [[1,1,1,2,2],[3]] => 1 [[1,1,1,2,3],[2]] => 0 [[1,1,1,2,3],[3]] => 1 [[1,1,1,3,3],[2]] => 0 [[1,1,1,3,3],[3]] => 1 [[1,1,2,2,2],[3]] => 1 [[1,1,2,2,3],[2]] => 0 [[1,1,2,2,3],[3]] => 1 [[1,1,2,3,3],[2]] => 0 [[1,1,2,3,3],[3]] => 1 [[1,1,3,3,3],[2]] => 0 [[1,1,3,3,3],[3]] => 1 [[1,2,2,2,2],[3]] => 1 [[1,2,2,2,3],[2]] => 1 [[1,2,2,2,3],[3]] => 1 [[1,2,2,3,3],[2]] => 1 [[1,2,2,3,3],[3]] => 1 [[1,2,3,3,3],[2]] => 1 [[1,2,3,3,3],[3]] => 1 [[1,3,3,3,3],[2]] => 1 [[1,3,3,3,3],[3]] => 2 [[2,2,2,2,2],[3]] => 2 [[2,2,2,2,3],[3]] => 2 [[2,2,2,3,3],[3]] => 2 [[2,2,3,3,3],[3]] => 2 [[2,3,3,3,3],[3]] => 3 [[1,1,1,1],[2,3]] => 0 [[1,1,1,1],[3,3]] => 1 [[1,1,1,2],[2,3]] => 0 [[1,1,1,3],[2,2]] => 0 [[1,1,1,2],[3,3]] => 1 [[1,1,1,3],[2,3]] => 0 [[1,1,1,3],[3,3]] => 1 [[1,1,2,2],[2,3]] => 0 [[1,1,2,3],[2,2]] => 1 [[1,1,2,2],[3,3]] => 1 [[1,1,2,3],[2,3]] => 0 [[1,1,3,3],[2,2]] => 1 [[1,1,2,3],[3,3]] => 1 [[1,1,3,3],[2,3]] => 1 [[1,1,3,3],[3,3]] => 2 [[1,2,2,2],[2,3]] => 1 [[1,2,2,2],[3,3]] => 1 [[1,2,2,3],[2,3]] => 1 [[1,2,2,3],[3,3]] => 1 [[1,2,3,3],[2,3]] => 2 [[1,2,3,3],[3,3]] => 2 [[2,2,2,2],[3,3]] => 2 [[2,2,2,3],[3,3]] => 2 [[2,2,3,3],[3,3]] => 3 [[1,1,1,1],[2],[3]] => 0 [[1,1,1,2],[2],[3]] => 0 [[1,1,1,3],[2],[3]] => 0 [[1,1,2,2],[2],[3]] => 0 [[1,1,2,3],[2],[3]] => 0 [[1,1,3,3],[2],[3]] => 0 [[1,2,2,2],[2],[3]] => 1 [[1,2,2,3],[2],[3]] => 1 [[1,2,3,3],[2],[3]] => 1 [[1,3,3,3],[2],[3]] => 1 [[1,1,1],[2,2,3]] => 0 [[1,1,1],[2,3,3]] => 0 [[1,1,1],[3,3,3]] => 1 [[1,1,2],[2,2,3]] => 1 [[1,1,2],[2,3,3]] => 0 [[1,1,2],[3,3,3]] => 1 [[1,2,2],[2,3,3]] => 1 [[1,2,2],[3,3,3]] => 1 [[2,2,2],[3,3,3]] => 2 [[1,1,1],[2,2],[3]] => 0 [[1,1,1],[2,3],[3]] => 1 [[1,1,2],[2,2],[3]] => 1 [[1,1,2],[2,3],[3]] => 1 [[1,1,3],[2,2],[3]] => 1 [[1,1,3],[2,3],[3]] => 2 [[1,2,2],[2,3],[3]] => 2 [[1,2,3],[2,3],[3]] => 3 [[1,1],[2,2],[3,3]] => 0 [[1,1,1,1,1,4]] => 0 [[1,1,1,1,2,4]] => 0 [[1,1,1,1,3,4]] => 0 [[1,1,1,1,4,4]] => 0 [[1,1,1,2,2,4]] => 0 [[1,1,1,2,3,4]] => 0 [[1,1,1,2,4,4]] => 0 [[1,1,1,3,3,4]] => 0 [[1,1,1,3,4,4]] => 0 [[1,1,1,4,4,4]] => 0 [[1,1,2,2,2,4]] => 0 [[1,1,2,2,3,4]] => 0 [[1,1,2,2,4,4]] => 0 [[1,1,2,3,3,4]] => 0 [[1,1,2,3,4,4]] => 0 [[1,1,2,4,4,4]] => 0 [[1,1,3,3,3,4]] => 0 [[1,1,3,3,4,4]] => 0 [[1,1,3,4,4,4]] => 0 [[1,1,4,4,4,4]] => 0 [[1,2,2,2,2,4]] => 0 [[1,2,2,2,3,4]] => 0 [[1,2,2,2,4,4]] => 0 [[1,2,2,3,3,4]] => 0 [[1,2,2,3,4,4]] => 0 [[1,2,2,4,4,4]] => 0 [[1,2,3,3,3,4]] => 0 [[1,2,3,3,4,4]] => 0 [[1,2,3,4,4,4]] => 0 [[1,2,4,4,4,4]] => 0 [[1,3,3,3,3,4]] => 0 [[1,3,3,3,4,4]] => 0 [[1,3,3,4,4,4]] => 0 [[1,3,4,4,4,4]] => 0 [[1,4,4,4,4,4]] => 0 [[2,2,2,2,2,4]] => 1 [[2,2,2,2,3,4]] => 1 [[2,2,2,2,4,4]] => 1 [[2,2,2,3,3,4]] => 1 [[2,2,2,3,4,4]] => 1 [[2,2,2,4,4,4]] => 1 [[2,2,3,3,3,4]] => 1 [[2,2,3,3,4,4]] => 1 [[2,2,3,4,4,4]] => 1 [[2,2,4,4,4,4]] => 1 [[2,3,3,3,3,4]] => 1 [[2,3,3,3,4,4]] => 1 [[2,3,3,4,4,4]] => 1 [[2,3,4,4,4,4]] => 1 [[2,4,4,4,4,4]] => 1 [[3,3,3,3,3,4]] => 1 [[3,3,3,3,4,4]] => 1 [[3,3,3,4,4,4]] => 1 [[3,3,4,4,4,4]] => 1 [[3,4,4,4,4,4]] => 1 [[4,4,4,4,4,4]] => 1 [[1,1,1,1,1],[4]] => 1 [[1,1,1,1,2],[4]] => 1 [[1,1,1,1,4],[2]] => 0 [[1,1,1,1,3],[4]] => 1 [[1,1,1,1,4],[3]] => 1 [[1,1,1,1,4],[4]] => 1 [[1,1,1,2,2],[4]] => 1 [[1,1,1,2,4],[2]] => 0 [[1,1,1,2,3],[4]] => 1 [[1,1,1,2,4],[3]] => 1 [[1,1,1,3,4],[2]] => 0 [[1,1,1,2,4],[4]] => 1 [[1,1,1,4,4],[2]] => 0 [[1,1,1,3,3],[4]] => 1 [[1,1,1,3,4],[3]] => 1 [[1,1,1,3,4],[4]] => 1 [[1,1,1,4,4],[3]] => 1 [[1,1,1,4,4],[4]] => 1 [[1,1,2,2,2],[4]] => 1 [[1,1,2,2,4],[2]] => 0 [[1,1,2,2,3],[4]] => 1 [[1,1,2,2,4],[3]] => 1 [[1,1,2,3,4],[2]] => 0 [[1,1,2,2,4],[4]] => 1 [[1,1,2,4,4],[2]] => 0 [[1,1,2,3,3],[4]] => 1 [[1,1,2,3,4],[3]] => 1 [[1,1,3,3,4],[2]] => 0 [[1,1,2,3,4],[4]] => 1 [[1,1,2,4,4],[3]] => 1 [[1,1,3,4,4],[2]] => 0 [[1,1,2,4,4],[4]] => 1 [[1,1,4,4,4],[2]] => 0 [[1,1,3,3,3],[4]] => 1 [[1,1,3,3,4],[3]] => 1 [[1,1,3,3,4],[4]] => 1 [[1,1,3,4,4],[3]] => 1 [[1,1,3,4,4],[4]] => 1 [[1,1,4,4,4],[3]] => 1 [[1,1,4,4,4],[4]] => 1 [[1,2,2,2,2],[4]] => 1 [[1,2,2,2,4],[2]] => 1 [[1,2,2,2,3],[4]] => 1 [[1,2,2,2,4],[3]] => 1 [[1,2,2,3,4],[2]] => 1 [[1,2,2,2,4],[4]] => 1 [[1,2,2,4,4],[2]] => 1 [[1,2,2,3,3],[4]] => 1 [[1,2,2,3,4],[3]] => 1 [[1,2,3,3,4],[2]] => 1 [[1,2,2,3,4],[4]] => 1 [[1,2,2,4,4],[3]] => 1 [[1,2,3,4,4],[2]] => 1 [[1,2,2,4,4],[4]] => 1 [[1,2,4,4,4],[2]] => 1 [[1,2,3,3,3],[4]] => 1 [[1,2,3,3,4],[3]] => 1 [[1,3,3,3,4],[2]] => 1 [[1,2,3,3,4],[4]] => 1 [[1,2,3,4,4],[3]] => 1 [[1,3,3,4,4],[2]] => 1 [[1,2,3,4,4],[4]] => 1 [[1,2,4,4,4],[3]] => 1 [[1,3,4,4,4],[2]] => 1 [[1,2,4,4,4],[4]] => 1 [[1,4,4,4,4],[2]] => 1 [[1,3,3,3,3],[4]] => 1 [[1,3,3,3,4],[3]] => 2 [[1,3,3,3,4],[4]] => 1 [[1,3,3,4,4],[3]] => 2 [[1,3,3,4,4],[4]] => 1 [[1,3,4,4,4],[3]] => 2 [[1,3,4,4,4],[4]] => 1 [[1,4,4,4,4],[3]] => 2 [[1,4,4,4,4],[4]] => 2 [[2,2,2,2,2],[4]] => 2 [[2,2,2,2,3],[4]] => 2 [[2,2,2,2,4],[3]] => 2 [[2,2,2,2,4],[4]] => 2 [[2,2,2,3,3],[4]] => 2 [[2,2,2,3,4],[3]] => 2 [[2,2,2,3,4],[4]] => 2 [[2,2,2,4,4],[3]] => 2 [[2,2,2,4,4],[4]] => 2 [[2,2,3,3,3],[4]] => 2 [[2,2,3,3,4],[3]] => 2 [[2,2,3,3,4],[4]] => 2 [[2,2,3,4,4],[3]] => 2 [[2,2,3,4,4],[4]] => 2 [[2,2,4,4,4],[3]] => 2 [[2,2,4,4,4],[4]] => 2 [[2,3,3,3,3],[4]] => 2 [[2,3,3,3,4],[3]] => 3 [[2,3,3,3,4],[4]] => 2 [[2,3,3,4,4],[3]] => 3 [[2,3,3,4,4],[4]] => 2 [[2,3,4,4,4],[3]] => 3 [[2,3,4,4,4],[4]] => 2 [[2,4,4,4,4],[3]] => 3 [[2,4,4,4,4],[4]] => 3 [[3,3,3,3,3],[4]] => 2 [[3,3,3,3,4],[4]] => 2 [[3,3,3,4,4],[4]] => 2 [[3,3,4,4,4],[4]] => 2 [[3,4,4,4,4],[4]] => 3 [[1,1,1,1],[2,4]] => 0 [[1,1,1,1],[3,4]] => 1 [[1,1,1,1],[4,4]] => 1 [[1,1,1,2],[2,4]] => 0 [[1,1,1,4],[2,2]] => 0 [[1,1,1,2],[3,4]] => 1 [[1,1,1,3],[2,4]] => 0 [[1,1,1,4],[2,3]] => 0 [[1,1,1,2],[4,4]] => 1 [[1,1,1,4],[2,4]] => 0 [[1,1,1,3],[3,4]] => 1 [[1,1,1,4],[3,3]] => 1 [[1,1,1,3],[4,4]] => 1 [[1,1,1,4],[3,4]] => 1 [[1,1,1,4],[4,4]] => 1 [[1,1,2,2],[2,4]] => 0 [[1,1,2,4],[2,2]] => 1 [[1,1,2,2],[3,4]] => 1 [[1,1,2,3],[2,4]] => 0 [[1,1,2,4],[2,3]] => 0 [[1,1,3,4],[2,2]] => 1 [[1,1,2,2],[4,4]] => 1 [[1,1,2,4],[2,4]] => 0 [[1,1,4,4],[2,2]] => 1 [[1,1,2,3],[3,4]] => 1 [[1,1,2,4],[3,3]] => 1 [[1,1,3,3],[2,4]] => 0 [[1,1,3,4],[2,3]] => 1 [[1,1,2,3],[4,4]] => 1 [[1,1,2,4],[3,4]] => 1 [[1,1,3,4],[2,4]] => 0 [[1,1,4,4],[2,3]] => 1 [[1,1,2,4],[4,4]] => 1 [[1,1,4,4],[2,4]] => 1 [[1,1,3,3],[3,4]] => 1 [[1,1,3,4],[3,3]] => 2 [[1,1,3,3],[4,4]] => 1 [[1,1,3,4],[3,4]] => 1 [[1,1,4,4],[3,3]] => 2 [[1,1,3,4],[4,4]] => 1 [[1,1,4,4],[3,4]] => 2 [[1,1,4,4],[4,4]] => 2 [[1,2,2,2],[2,4]] => 1 [[1,2,2,2],[3,4]] => 2 [[1,2,2,3],[2,4]] => 1 [[1,2,2,4],[2,3]] => 1 [[1,2,2,2],[4,4]] => 1 [[1,2,2,4],[2,4]] => 1 [[1,2,2,3],[3,4]] => 2 [[1,2,2,4],[3,3]] => 1 [[1,2,3,3],[2,4]] => 1 [[1,2,3,4],[2,3]] => 2 [[1,2,2,3],[4,4]] => 1 [[1,2,2,4],[3,4]] => 2 [[1,2,3,4],[2,4]] => 1 [[1,2,4,4],[2,3]] => 2 [[1,2,2,4],[4,4]] => 1 [[1,2,4,4],[2,4]] => 2 [[1,2,3,3],[3,4]] => 2 [[1,2,3,4],[3,3]] => 2 [[1,3,3,3],[2,4]] => 1 [[1,2,3,3],[4,4]] => 1 [[1,2,3,4],[3,4]] => 2 [[1,2,4,4],[3,3]] => 2 [[1,3,3,4],[2,4]] => 1 [[1,2,3,4],[4,4]] => 1 [[1,2,4,4],[3,4]] => 3 [[1,3,4,4],[2,4]] => 2 [[1,2,4,4],[4,4]] => 2 [[1,3,3,3],[3,4]] => 2 [[1,3,3,3],[4,4]] => 1 [[1,3,3,4],[3,4]] => 2 [[1,3,3,4],[4,4]] => 1 [[1,3,4,4],[3,4]] => 3 [[1,3,4,4],[4,4]] => 2 [[2,2,2,2],[3,4]] => 2 [[2,2,2,2],[4,4]] => 2 [[2,2,2,3],[3,4]] => 2 [[2,2,2,4],[3,3]] => 2 [[2,2,2,3],[4,4]] => 2 [[2,2,2,4],[3,4]] => 2 [[2,2,2,4],[4,4]] => 2 [[2,2,3,3],[3,4]] => 2 [[2,2,3,4],[3,3]] => 3 [[2,2,3,3],[4,4]] => 2 [[2,2,3,4],[3,4]] => 2 [[2,2,4,4],[3,3]] => 3 [[2,2,3,4],[4,4]] => 2 [[2,2,4,4],[3,4]] => 3 [[2,2,4,4],[4,4]] => 3 [[2,3,3,3],[3,4]] => 3 [[2,3,3,3],[4,4]] => 2 [[2,3,3,4],[3,4]] => 3 [[2,3,3,4],[4,4]] => 2 [[2,3,4,4],[3,4]] => 4 [[2,3,4,4],[4,4]] => 3 [[3,3,3,3],[4,4]] => 2 [[3,3,3,4],[4,4]] => 2 [[3,3,4,4],[4,4]] => 3 [[1,1,1,1],[2],[4]] => 1 [[1,1,1,1],[3],[4]] => 2 [[1,1,1,2],[2],[4]] => 1 [[1,1,1,2],[3],[4]] => 2 [[1,1,1,3],[2],[4]] => 1 [[1,1,1,4],[2],[3]] => 0 [[1,1,1,4],[2],[4]] => 1 [[1,1,1,3],[3],[4]] => 2 [[1,1,1,4],[3],[4]] => 2 [[1,1,2,2],[2],[4]] => 1 [[1,1,2,2],[3],[4]] => 2 [[1,1,2,3],[2],[4]] => 1 [[1,1,2,4],[2],[3]] => 0 [[1,1,2,4],[2],[4]] => 1 [[1,1,2,3],[3],[4]] => 2 [[1,1,3,3],[2],[4]] => 1 [[1,1,3,4],[2],[3]] => 0 [[1,1,2,4],[3],[4]] => 2 [[1,1,3,4],[2],[4]] => 1 [[1,1,4,4],[2],[3]] => 0 [[1,1,4,4],[2],[4]] => 1 [[1,1,3,3],[3],[4]] => 2 [[1,1,3,4],[3],[4]] => 2 [[1,1,4,4],[3],[4]] => 2 [[1,2,2,2],[2],[4]] => 2 [[1,2,2,2],[3],[4]] => 2 [[1,2,2,3],[2],[4]] => 2 [[1,2,2,4],[2],[3]] => 1 [[1,2,2,4],[2],[4]] => 2 [[1,2,2,3],[3],[4]] => 2 [[1,2,3,3],[2],[4]] => 2 [[1,2,3,4],[2],[3]] => 1 [[1,2,2,4],[3],[4]] => 2 [[1,2,3,4],[2],[4]] => 2 [[1,2,4,4],[2],[3]] => 1 [[1,2,4,4],[2],[4]] => 2 [[1,2,3,3],[3],[4]] => 2 [[1,3,3,3],[2],[4]] => 2 [[1,3,3,4],[2],[3]] => 1 [[1,2,3,4],[3],[4]] => 2 [[1,3,3,4],[2],[4]] => 2 [[1,3,4,4],[2],[3]] => 1 [[1,2,4,4],[3],[4]] => 2 [[1,3,4,4],[2],[4]] => 2 [[1,4,4,4],[2],[3]] => 1 [[1,4,4,4],[2],[4]] => 2 [[1,3,3,3],[3],[4]] => 3 [[1,3,3,4],[3],[4]] => 3 [[1,3,4,4],[3],[4]] => 3 [[1,4,4,4],[3],[4]] => 3 [[2,2,2,2],[3],[4]] => 3 [[2,2,2,3],[3],[4]] => 3 [[2,2,2,4],[3],[4]] => 3 [[2,2,3,3],[3],[4]] => 3 [[2,2,3,4],[3],[4]] => 3 [[2,2,4,4],[3],[4]] => 3 [[2,3,3,3],[3],[4]] => 4 [[2,3,3,4],[3],[4]] => 4 [[2,3,4,4],[3],[4]] => 4 [[2,4,4,4],[3],[4]] => 4 [[1,1,1],[2,2,4]] => 0 [[1,1,1],[2,3,4]] => 0 [[1,1,1],[2,4,4]] => 0 [[1,1,1],[3,3,4]] => 1 [[1,1,1],[3,4,4]] => 1 [[1,1,1],[4,4,4]] => 1 [[1,1,2],[2,2,4]] => 1 [[1,1,2],[2,3,4]] => 1 [[1,1,3],[2,2,4]] => 1 [[1,1,2],[2,4,4]] => 0 [[1,1,2],[3,3,4]] => 2 [[1,1,3],[2,3,4]] => 1 [[1,1,2],[3,4,4]] => 1 [[1,1,3],[2,4,4]] => 0 [[1,1,2],[4,4,4]] => 1 [[1,1,3],[3,3,4]] => 2 [[1,1,3],[3,4,4]] => 1 [[1,1,3],[4,4,4]] => 1 [[1,2,2],[2,3,4]] => 1 [[1,2,2],[2,4,4]] => 1 [[1,2,2],[3,3,4]] => 1 [[1,2,3],[2,3,4]] => 2 [[1,2,2],[3,4,4]] => 2 [[1,2,3],[2,4,4]] => 1 [[1,2,2],[4,4,4]] => 1 [[1,2,3],[3,3,4]] => 2 [[1,2,3],[3,4,4]] => 2 [[1,3,3],[2,4,4]] => 1 [[1,2,3],[4,4,4]] => 1 [[1,3,3],[3,4,4]] => 2 [[1,3,3],[4,4,4]] => 1 [[2,2,2],[3,3,4]] => 2 [[2,2,2],[3,4,4]] => 2 [[2,2,2],[4,4,4]] => 2 [[2,2,3],[3,3,4]] => 3 [[2,2,3],[3,4,4]] => 2 [[2,2,3],[4,4,4]] => 2 [[2,3,3],[3,4,4]] => 3 [[2,3,3],[4,4,4]] => 2 [[3,3,3],[4,4,4]] => 2 [[1,1,1],[2,2],[4]] => 1 [[1,1,1],[2,3],[4]] => 1 [[1,1,1],[2,4],[3]] => 1 [[1,1,1],[2,4],[4]] => 2 [[1,1,1],[3,3],[4]] => 2 [[1,1,1],[3,4],[4]] => 3 [[1,1,2],[2,2],[4]] => 2 [[1,1,2],[2,3],[4]] => 1 [[1,1,2],[2,4],[3]] => 1 [[1,1,3],[2,2],[4]] => 2 [[1,1,4],[2,2],[3]] => 1 [[1,1,2],[2,4],[4]] => 2 [[1,1,4],[2,2],[4]] => 2 [[1,1,2],[3,3],[4]] => 2 [[1,1,3],[2,3],[4]] => 2 [[1,1,3],[2,4],[3]] => 1 [[1,1,4],[2,3],[3]] => 2 [[1,1,2],[3,4],[4]] => 3 [[1,1,3],[2,4],[4]] => 2 [[1,1,4],[2,3],[4]] => 2 [[1,1,4],[2,4],[3]] => 2 [[1,1,4],[2,4],[4]] => 3 [[1,1,3],[3,3],[4]] => 3 [[1,1,3],[3,4],[4]] => 3 [[1,1,4],[3,3],[4]] => 3 [[1,1,4],[3,4],[4]] => 4 [[1,2,2],[2,3],[4]] => 2 [[1,2,2],[2,4],[3]] => 2 [[1,2,2],[2,4],[4]] => 3 [[1,2,2],[3,3],[4]] => 2 [[1,2,3],[2,3],[4]] => 3 [[1,2,3],[2,4],[3]] => 2 [[1,2,4],[2,3],[3]] => 3 [[1,2,2],[3,4],[4]] => 4 [[1,2,3],[2,4],[4]] => 3 [[1,2,4],[2,3],[4]] => 3 [[1,2,4],[2,4],[3]] => 3 [[1,2,4],[2,4],[4]] => 4 [[1,2,3],[3,3],[4]] => 3 [[1,3,3],[2,4],[3]] => 2 [[1,2,3],[3,4],[4]] => 4 [[1,2,4],[3,3],[4]] => 3 [[1,3,3],[2,4],[4]] => 3 [[1,3,4],[2,4],[3]] => 3 [[1,2,4],[3,4],[4]] => 5 [[1,3,4],[2,4],[4]] => 4 [[1,3,3],[3,4],[4]] => 4 [[1,3,4],[3,4],[4]] => 5 [[2,2,2],[3,3],[4]] => 3 [[2,2,2],[3,4],[4]] => 4 [[2,2,3],[3,3],[4]] => 4 [[2,2,3],[3,4],[4]] => 4 [[2,2,4],[3,3],[4]] => 4 [[2,2,4],[3,4],[4]] => 5 [[2,3,3],[3,4],[4]] => 5 [[2,3,4],[3,4],[4]] => 6 [[1,1,1],[2],[3],[4]] => 0 [[1,1,2],[2],[3],[4]] => 0 [[1,1,3],[2],[3],[4]] => 0 [[1,1,4],[2],[3],[4]] => 0 [[1,2,2],[2],[3],[4]] => 1 [[1,2,3],[2],[3],[4]] => 1 [[1,2,4],[2],[3],[4]] => 1 [[1,3,3],[2],[3],[4]] => 1 [[1,3,4],[2],[3],[4]] => 1 [[1,4,4],[2],[3],[4]] => 1 [[1,1],[2,2],[3,4]] => 0 [[1,1],[2,2],[4,4]] => 1 [[1,1],[2,3],[3,4]] => 1 [[1,1],[2,3],[4,4]] => 1 [[1,1],[3,3],[4,4]] => 2 [[1,2],[2,3],[3,4]] => 2 [[1,2],[2,3],[4,4]] => 2 [[1,2],[3,3],[4,4]] => 2 [[2,2],[3,3],[4,4]] => 3 [[1,1],[2,2],[3],[4]] => 0 [[1,1],[2,3],[3],[4]] => 1 [[1,1],[2,4],[3],[4]] => 1 [[1,2],[2,3],[3],[4]] => 2 [[1,2],[2,4],[3],[4]] => 2 [[1,3],[2,4],[3],[4]] => 2 ----------------------------------------------------------------------------- Created: Jan 15, 2014 at 19:13 by Ben Salisbury ----------------------------------------------------------------------------- Last Updated: Oct 19, 2015 at 16:34 by Christian Stump