Identifier
-
Mp00001:
Alternating sign matrices
—to semistandard tableau via monotone triangles⟶
Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤ (values match St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: )
Values
[[1]] => [[1]] => [1] => [1,0,1,0] => 1
[[1,0],[0,1]] => [[1,1],[2]] => [2,1] => [1,0,1,0,1,0] => 2
[[0,1],[1,0]] => [[1,2],[2]] => [2,1] => [1,0,1,0,1,0] => 2
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 1
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 3
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[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]] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[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]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,3,5],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,2,3],[2,2,3,3],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,2,2],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,2,5],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,3,3],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]] => [[1,1,2,3,5],[2,2,3,5],[3,3,5],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]] => [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
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Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the bounce path of $D$.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the bounce path of $D$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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.
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
weight
Description
The weight of a semistandard tableau as an integer partition.
The weight (or content) of a semistandard tableaux $T$ with maximal entry $m$ is the weak composition $(\alpha_1, \dots, \alpha_m)$ such that $\alpha_i$ is the number of letters $i$ occurring in $T$.
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.
The weight (or content) of a semistandard tableaux $T$ with maximal entry $m$ is the weak composition $(\alpha_1, \dots, \alpha_m)$ such that $\alpha_i$ is the number of letters $i$ occurring in $T$.
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.
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