Identifier
-
Mp00001:
Alternating sign matrices
—to semistandard tableau via monotone triangles⟶
Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000478: Integer partitions ⟶ ℤ
Values
[[1,0],[0,1]] => [[1,1],[2]] => [2,1] => 0
[[0,1],[1,0]] => [[1,2],[2]] => [2,1] => 0
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [3,2,1] => 0
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [3,2,1] => 0
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [3,2,1] => 0
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [2,2,2] => 1
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [3,2,1] => 0
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [3,2,1] => 0
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [3,2,1] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 0
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Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
According to Theorem 3.4 (Alladi 2012) in [1]
$$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
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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