Identifier
Mp00001: to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00071: Permutations descent compositionInteger compositions
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1
[[1]]=>[[1]]=>[1]=>[1]=>[1] [[1,0],[0,1]]=>[[1,1],[2]]=>[3,1,2]=>[3,1,2]=>[1,2] [[0,1],[1,0]]=>[[1,2],[2]]=>[2,1,3]=>[2,1,3]=>[1,2] [[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[6,4,5,1,2,3]=>[6,4,5,1,3,2]=>[1,2,2,1] [[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[6,3,4,1,2,5]=>[6,3,5,1,4,2]=>[1,2,2,1] [[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[5,4,6,1,2,3]=>[5,4,6,1,3,2]=>[1,2,2,1] [[0,1,0],[1,-1,1],[0,1,0]]=>[[1,1,2],[2,3],[3]]=>[5,3,6,1,2,4]=>[5,3,6,1,4,2]=>[1,2,2,1] [[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[4,3,5,1,2,6]=>[4,3,6,1,5,2]=>[1,2,2,1] [[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[5,2,6,1,3,4]=>[5,2,6,1,4,3]=>[1,2,2,1] [[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[4,2,5,1,3,6]=>[4,2,6,1,5,3]=>[1,2,2,1] [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]=>[[1,1,1,1],[2,2,2],[3,3],[4]]=>[10,8,9,5,6,7,1,2,3,4]=>[10,8,9,5,7,6,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]=>[[1,1,1,2],[2,2,2],[3,3],[4]]=>[10,8,9,4,5,6,1,2,3,7]=>[10,8,9,4,7,6,1,5,3,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]=>[[1,1,1,1],[2,2,3],[3,3],[4]]=>[10,7,8,5,6,9,1,2,3,4]=>[10,7,9,5,8,6,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]=>[[1,1,1,2],[2,2,3],[3,3],[4]]=>[10,7,8,4,5,9,1,2,3,6]=>[10,7,9,4,8,6,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]=>[[1,1,1,3],[2,2,3],[3,3],[4]]=>[10,6,7,4,5,8,1,2,3,9]=>[10,6,9,4,8,7,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]=>[[1,1,2,2],[2,2,3],[3,3],[4]]=>[10,7,8,3,4,9,1,2,5,6]=>[10,7,9,3,8,6,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]=>[[1,1,2,3],[2,2,3],[3,3],[4]]=>[10,6,7,3,4,8,1,2,5,9]=>[10,6,9,3,8,7,1,5,4,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]=>[[1,1,1,1],[2,2,2],[3,4],[4]]=>[9,8,10,5,6,7,1,2,3,4]=>[9,8,10,5,7,6,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]=>[[1,1,1,2],[2,2,2],[3,4],[4]]=>[9,8,10,4,5,6,1,2,3,7]=>[9,8,10,4,7,6,1,5,3,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]=>[[1,1,1,1],[2,2,3],[3,4],[4]]=>[9,7,10,5,6,8,1,2,3,4]=>[9,7,10,5,8,6,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]=>[[1,1,1,2],[2,2,3],[3,4],[4]]=>[9,7,10,4,5,8,1,2,3,6]=>[9,7,10,4,8,6,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]=>[[1,1,1,3],[2,2,3],[3,4],[4]]=>[9,6,10,4,5,7,1,2,3,8]=>[9,6,10,4,8,7,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]=>[[1,1,2,2],[2,2,3],[3,4],[4]]=>[9,7,10,3,4,8,1,2,5,6]=>[9,7,10,3,8,6,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]=>[[1,1,2,3],[2,2,3],[3,4],[4]]=>[9,6,10,3,4,7,1,2,5,8]=>[9,6,10,3,8,7,1,5,4,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]=>[[1,1,1,1],[2,2,4],[3,4],[4]]=>[8,7,9,5,6,10,1,2,3,4]=>[8,7,10,5,9,6,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]=>[[1,1,1,2],[2,2,4],[3,4],[4]]=>[8,7,9,4,5,10,1,2,3,6]=>[8,7,10,4,9,6,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]=>[[1,1,1,3],[2,2,4],[3,4],[4]]=>[8,6,9,4,5,10,1,2,3,7]=>[8,6,10,4,9,7,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]=>[[1,1,1,4],[2,2,4],[3,4],[4]]=>[7,6,8,4,5,9,1,2,3,10]=>[7,6,10,4,9,8,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]=>[[1,1,2,2],[2,2,4],[3,4],[4]]=>[8,7,9,3,4,10,1,2,5,6]=>[8,7,10,3,9,6,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]=>[[1,1,2,3],[2,2,4],[3,4],[4]]=>[8,6,9,3,4,10,1,2,5,7]=>[8,6,10,3,9,7,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]=>[[1,1,2,4],[2,2,4],[3,4],[4]]=>[7,6,8,3,4,9,1,2,5,10]=>[7,6,10,3,9,8,1,5,4,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]=>[[1,1,1,1],[2,3,3],[3,4],[4]]=>[9,6,10,5,7,8,1,2,3,4]=>[9,6,10,5,8,7,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]=>[[1,1,1,2],[2,3,3],[3,4],[4]]=>[9,6,10,4,7,8,1,2,3,5]=>[9,6,10,4,8,7,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]=>[[1,1,1,3],[2,3,3],[3,4],[4]]=>[9,5,10,4,6,7,1,2,3,8]=>[9,5,10,4,8,7,1,6,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]=>[[1,1,2,2],[2,3,3],[3,4],[4]]=>[9,6,10,3,7,8,1,2,4,5]=>[9,6,10,3,8,7,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]=>[[1,1,2,3],[2,3,3],[3,4],[4]]=>[9,5,10,3,6,7,1,2,4,8]=>[9,5,10,3,8,7,1,6,4,2]=>[1,2,2,1,2,1,1] [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]=>[[1,1,1,1],[2,3,4],[3,4],[4]]=>[8,6,9,5,7,10,1,2,3,4]=>[8,6,10,5,9,7,1,4,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]=>[[1,1,1,2],[2,3,4],[3,4],[4]]=>[8,6,9,4,7,10,1,2,3,5]=>[8,6,10,4,9,7,1,5,3,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]=>[[1,1,1,3],[2,3,4],[3,4],[4]]=>[8,5,9,4,6,10,1,2,3,7]=>[8,5,10,4,9,7,1,6,3,2]=>[1,2,2,1,2,1,1] [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]=>[[1,1,1,4],[2,3,4],[3,4],[4]]=>[7,5,8,4,6,9,1,2,3,10]=>[7,5,10,4,9,8,1,6,3,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]=>[[1,1,2,2],[2,3,4],[3,4],[4]]=>[8,6,9,3,7,10,1,2,4,5]=>[8,6,10,3,9,7,1,5,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]=>[[1,1,2,3],[2,3,4],[3,4],[4]]=>[8,5,9,3,6,10,1,2,4,7]=>[8,5,10,3,9,7,1,6,4,2]=>[1,2,2,1,2,1,1] [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]=>[[1,1,2,4],[2,3,4],[3,4],[4]]=>[7,5,8,3,6,9,1,2,4,10]=>[7,5,10,3,9,8,1,6,4,2]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]=>[[1,1,3,3],[2,3,4],[3,4],[4]]=>[8,4,9,3,5,10,1,2,6,7]=>[8,4,10,3,9,7,1,6,5,2]=>[1,2,2,1,2,1,1] [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]=>[[1,1,3,4],[2,3,4],[3,4],[4]]=>[7,4,8,3,5,9,1,2,6,10]=>[7,4,10,3,9,8,1,6,5,2]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]=>[[1,2,2,2],[2,3,3],[3,4],[4]]=>[9,6,10,2,7,8,1,3,4,5]=>[9,6,10,2,8,7,1,5,4,3]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]=>[[1,2,2,3],[2,3,3],[3,4],[4]]=>[9,5,10,2,6,7,1,3,4,8]=>[9,5,10,2,8,7,1,6,4,3]=>[1,2,2,1,2,1,1] [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]=>[[1,2,2,2],[2,3,4],[3,4],[4]]=>[8,6,9,2,7,10,1,3,4,5]=>[8,6,10,2,9,7,1,5,4,3]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]=>[[1,2,2,3],[2,3,4],[3,4],[4]]=>[8,5,9,2,6,10,1,3,4,7]=>[8,5,10,2,9,7,1,6,4,3]=>[1,2,2,1,2,1,1] [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]=>[[1,2,2,4],[2,3,4],[3,4],[4]]=>[7,5,8,2,6,9,1,3,4,10]=>[7,5,10,2,9,8,1,6,4,3]=>[1,2,2,1,2,1,1] [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]=>[[1,2,3,3],[2,3,4],[3,4],[4]]=>[8,4,9,2,5,10,1,3,6,7]=>[8,4,10,2,9,7,1,6,5,3]=>[1,2,2,1,2,1,1] [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]=>[[1,2,3,4],[2,3,4],[3,4],[4]]=>[7,4,8,2,5,9,1,3,6,10]=>[7,4,10,2,9,8,1,6,5,3]=>[1,2,2,1,2,1,1]
Map
to semistandard tableau via monotone triangles
Description
The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.
Map
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.