Identifier
Mp00001: to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00075: reading word permutationPermutations
Mp00069: Permutations complementPermutations
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]=>[1,3,2]=>[2,1] [[0,1],[1,0]]=>[[1,2],[2]]=>[2,1,3]=>[2,3,1]=>[2,1] [[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[6,4,5,1,2,3]=>[1,3,2,6,5,4]=>[2,2,1,1] [[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[6,3,4,1,2,5]=>[1,4,3,6,5,2]=>[2,2,1,1] [[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[5,4,6,1,2,3]=>[2,3,1,6,5,4]=>[2,2,1,1] [[0,1,0],[1,-1,1],[0,1,0]]=>[[1,1,2],[2,3],[3]]=>[5,3,6,1,2,4]=>[2,4,1,6,5,3]=>[2,2,1,1] [[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[4,3,5,1,2,6]=>[3,4,2,6,5,1]=>[2,2,1,1] [[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[5,2,6,1,3,4]=>[2,5,1,6,4,3]=>[2,2,1,1] [[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[4,2,5,1,3,6]=>[3,5,2,6,4,1]=>[2,2,1,1] [[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]=>[1,3,2,6,5,4,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[1,3,2,7,6,5,10,9,8,4]=>[2,2,1,2,1,1,1] [[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]=>[1,4,3,6,5,2,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[1,4,3,7,6,2,10,9,8,5]=>[2,2,1,2,1,1,1] [[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]=>[1,5,4,7,6,3,10,9,8,2]=>[2,2,1,2,1,1,1] [[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]=>[1,4,3,8,7,2,10,9,6,5]=>[2,2,1,2,1,1,1] [[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]=>[1,5,4,8,7,3,10,9,6,2]=>[2,2,1,2,1,1,1] [[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]=>[2,3,1,6,5,4,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[2,3,1,7,6,5,10,9,8,4]=>[2,2,1,2,1,1,1] [[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]=>[2,4,1,6,5,3,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[2,4,1,7,6,3,10,9,8,5]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,7,6,4,10,9,8,3]=>[2,2,1,2,1,1,1] [[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]=>[2,4,1,8,7,3,10,9,6,5]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,8,7,4,10,9,6,3]=>[2,2,1,2,1,1,1] [[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]=>[3,4,2,6,5,1,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[3,4,2,7,6,1,10,9,8,5]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,7,6,1,10,9,8,4]=>[2,2,1,2,1,1,1] [[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]=>[4,5,3,7,6,2,10,9,8,1]=>[2,2,1,2,1,1,1] [[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]=>[3,4,2,8,7,1,10,9,6,5]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,8,7,1,10,9,6,4]=>[2,2,1,2,1,1,1] [[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]=>[4,5,3,8,7,2,10,9,6,1]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,6,4,3,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,7,4,3,10,9,8,6]=>[2,2,1,2,1,1,1] [[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]=>[2,6,1,7,5,4,10,9,8,3]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,8,4,3,10,9,7,6]=>[2,2,1,2,1,1,1] [[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]=>[2,6,1,8,5,4,10,9,7,3]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,6,4,1,10,9,8,7]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,7,4,1,10,9,8,6]=>[2,2,1,2,1,1,1] [[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]=>[3,6,2,7,5,1,10,9,8,4]=>[2,2,1,2,1,1,1] [[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]=>[4,6,3,7,5,2,10,9,8,1]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,8,4,1,10,9,7,6]=>[2,2,1,2,1,1,1] [[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]=>[3,6,2,8,5,1,10,9,7,4]=>[2,2,1,2,1,1,1] [[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]=>[4,6,3,8,5,2,10,9,7,1]=>[2,2,1,2,1,1,1] [[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]=>[3,7,2,8,6,1,10,9,5,4]=>[2,2,1,2,1,1,1] [[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]=>[4,7,3,8,6,2,10,9,5,1]=>[2,2,1,2,1,1,1] [[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]=>[2,5,1,9,4,3,10,8,7,6]=>[2,2,1,2,1,1,1] [[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]=>[2,6,1,9,5,4,10,8,7,3]=>[2,2,1,2,1,1,1] [[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]=>[3,5,2,9,4,1,10,8,7,6]=>[2,2,1,2,1,1,1] [[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]=>[3,6,2,9,5,1,10,8,7,4]=>[2,2,1,2,1,1,1] [[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]=>[4,6,3,9,5,2,10,8,7,1]=>[2,2,1,2,1,1,1] [[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]=>[3,7,2,9,6,1,10,8,5,4]=>[2,2,1,2,1,1,1] [[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]=>[4,7,3,9,6,2,10,8,5,1]=>[2,2,1,2,1,1,1]
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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
descent composition
Description
The descent composition of a permutation.
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.