- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001667: Permutations ⟶ ℤ

Values

[[1,2]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2,2]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1

[[1],[2]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[1,3]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2,3]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[3,3]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1

[[1],[3]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2],[3]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[1,1,2]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2,2]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,2,2]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1

[[1,1],[2]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2],[2]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2,4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[3,4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[4,4]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1

[[1],[4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2],[4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[3],[4]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[1,1,3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2,3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,3,3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,2,3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,3,3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[3,3,3]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1

[[1,1],[3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,3],[2]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,3],[3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,2],[3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,3],[3]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1],[2],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,1,1,2]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1,2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,2,2,2]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[2,2,2,2]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 2

[[1,1,1],[2]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1,2],[2]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,2,2],[2]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2,5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[3,5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[4,5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[5,5]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1

[[1],[5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[2],[5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[3],[5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[4],[5]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1

[[1,1,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2,4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,3,4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,4,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,2,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,3,4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[2,4,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[3,3,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[3,4,4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[4,4,4]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1

[[1,1],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1,2],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,4],[2]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,3],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,4],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,4],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,2],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[2,3],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[2,4],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[2,4],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[3,3],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[3,4],[4]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1

[[1],[2],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1],[3],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[2],[3],[4]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1

[[1,1,1,3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1,2,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,1,3,3]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,2,2,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,2,3,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,3,3,3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[2,2,2,3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[2,2,3,3]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[2,3,3,3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[3,3,3,3]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 2

[[1,1,1],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1,2],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,1,3],[2]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,1,3],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,2,2],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,2,3],[2]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,2,3],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,3,3],[2]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,3,3],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[2,2,2],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[2,2,3],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[2,3,3],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2

[[1,1],[2,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,1],[3,3]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2

[[1,2],[2,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

[[1,2],[3,3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2

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Description

The maximal size of a pair of weak twins for a permutation.
A pair of weak twins in a permutation is a pair of two disjoint subsequences of the same length with the same descent pattern. More formally, a pair of weak twins of size $k$ for a permutation $\pi$ of length $n$ are two disjoint lists $1 \leq i_1 < \dots < i_k \leq n$ and $1 \leq j_1 < \dots < j_k \leq n$ such that $\pi(i_a) < \pi(i_{a+1})$ if and only if $\pi(j_a) < \pi(j_{a+1})$ for all $1 \leq a < k$.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

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.