- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000900: Integer compositions ⟶ ℤ

Values

[[1],[]] => [] => => [1] => 1

[[2],[]] => [] => => [1] => 1

[[1,1],[]] => [] => => [1] => 1

[[2,1],[1]] => [1] => 10 => [1,2] => 1

[[3],[]] => [] => => [1] => 1

[[2,1],[]] => [] => => [1] => 1

[[3,1],[1]] => [1] => 10 => [1,2] => 1

[[2,2],[1]] => [1] => 10 => [1,2] => 1

[[3,2],[2]] => [2] => 100 => [1,3] => 1

[[1,1,1],[]] => [] => => [1] => 1

[[2,2,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[2,1,1],[1]] => [1] => 10 => [1,2] => 1

[[3,2,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4],[]] => [] => => [1] => 1

[[3,1],[]] => [] => => [1] => 1

[[4,1],[1]] => [1] => 10 => [1,2] => 1

[[2,2],[]] => [] => => [1] => 1

[[3,2],[1]] => [1] => 10 => [1,2] => 1

[[4,2],[2]] => [2] => 100 => [1,3] => 1

[[2,1,1],[]] => [] => => [1] => 1

[[3,2,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[3,1,1],[1]] => [1] => 10 => [1,2] => 1

[[4,2,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[3,3],[2]] => [2] => 100 => [1,3] => 1

[[4,3],[3]] => [3] => 1000 => [1,4] => 1

[[2,2,1],[1]] => [1] => 10 => [1,2] => 1

[[3,3,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[3,2,1],[2]] => [2] => 100 => [1,3] => 1

[[4,3,1],[3,1]] => [3,1] => 10010 => [1,3,2] => 1

[[2,2,2],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[3,3,2],[2,2]] => [2,2] => 1100 => [1,1,3] => 1

[[3,2,2],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4,3,2],[3,2]] => [3,2] => 10100 => [1,2,3] => 1

[[1,1,1,1],[]] => [] => => [1] => 1

[[2,2,2,1],[1,1,1]] => [1,1,1] => 1110 => [1,1,1,2] => 1

[[2,2,1,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[3,3,2,1],[2,2,1]] => [2,2,1] => 11010 => [1,1,2,2] => 2

[[2,1,1,1],[1]] => [1] => 10 => [1,2] => 1

[[3,2,2,1],[2,1,1]] => [2,1,1] => 10110 => [1,2,1,2] => 1

[[3,2,1,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4,3,2,1],[3,2,1]] => [3,2,1] => 101010 => [1,2,2,2] => 1

[[5],[]] => [] => => [1] => 1

[[4,1],[]] => [] => => [1] => 1

[[5,1],[1]] => [1] => 10 => [1,2] => 1

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

[[4,2],[1]] => [1] => 10 => [1,2] => 1

[[5,2],[2]] => [2] => 100 => [1,3] => 1

[[3,1,1],[]] => [] => => [1] => 1

[[4,2,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[4,1,1],[1]] => [1] => 10 => [1,2] => 1

[[5,2,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[3,3],[1]] => [1] => 10 => [1,2] => 1

[[4,3],[2]] => [2] => 100 => [1,3] => 1

[[5,3],[3]] => [3] => 1000 => [1,4] => 1

[[2,2,1],[]] => [] => => [1] => 1

[[3,3,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[3,2,1],[1]] => [1] => 10 => [1,2] => 1

[[4,3,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4,2,1],[2]] => [2] => 100 => [1,3] => 1

[[5,3,1],[3,1]] => [3,1] => 10010 => [1,3,2] => 1

[[3,2,2],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[4,3,2],[2,2]] => [2,2] => 1100 => [1,1,3] => 1

[[4,2,2],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[5,3,2],[3,2]] => [3,2] => 10100 => [1,2,3] => 1

[[2,1,1,1],[]] => [] => => [1] => 1

[[3,2,2,1],[1,1,1]] => [1,1,1] => 1110 => [1,1,1,2] => 1

[[3,2,1,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[4,3,2,1],[2,2,1]] => [2,2,1] => 11010 => [1,1,2,2] => 2

[[3,1,1,1],[1]] => [1] => 10 => [1,2] => 1

[[4,2,2,1],[2,1,1]] => [2,1,1] => 10110 => [1,2,1,2] => 1

[[4,2,1,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[5,3,2,1],[3,2,1]] => [3,2,1] => 101010 => [1,2,2,2] => 1

[[4,4],[3]] => [3] => 1000 => [1,4] => 1

[[5,4],[4]] => [4] => 10000 => [1,5] => 1

[[3,3,1],[2]] => [2] => 100 => [1,3] => 1

[[4,4,1],[3,1]] => [3,1] => 10010 => [1,3,2] => 1

[[4,3,1],[3]] => [3] => 1000 => [1,4] => 1

[[5,4,1],[4,1]] => [4,1] => 100010 => [1,4,2] => 1

[[2,2,2],[1]] => [1] => 10 => [1,2] => 1

[[3,3,2],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4,4,2],[3,2]] => [3,2] => 10100 => [1,2,3] => 1

[[3,2,2],[2]] => [2] => 100 => [1,3] => 1

[[4,3,2],[3,1]] => [3,1] => 10010 => [1,3,2] => 1

[[5,4,2],[4,2]] => [4,2] => 100100 => [1,3,3] => 1

[[2,2,1,1],[1]] => [1] => 10 => [1,2] => 1

[[3,3,2,1],[2,1,1]] => [2,1,1] => 10110 => [1,2,1,2] => 1

[[3,3,1,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

[[4,4,2,1],[3,2,1]] => [3,2,1] => 101010 => [1,2,2,2] => 1

[[3,2,1,1],[2]] => [2] => 100 => [1,3] => 1

[[4,3,2,1],[3,1,1]] => [3,1,1] => 100110 => [1,3,1,2] => 1

[[4,3,1,1],[3,1]] => [3,1] => 10010 => [1,3,2] => 1

[[5,4,2,1],[4,2,1]] => [4,2,1] => 1001010 => [1,3,2,2] => 1

[[3,3,3],[2,2]] => [2,2] => 1100 => [1,1,3] => 1

[[4,4,3],[3,3]] => [3,3] => 11000 => [1,1,4] => 1

[[4,3,3],[3,2]] => [3,2] => 10100 => [1,2,3] => 1

[[5,4,3],[4,3]] => [4,3] => 101000 => [1,2,4] => 1

[[2,2,2,1],[1,1]] => [1,1] => 110 => [1,1,2] => 1

[[3,3,3,1],[2,2,1]] => [2,2,1] => 11010 => [1,1,2,2] => 2

[[3,3,2,1],[2,2]] => [2,2] => 1100 => [1,1,3] => 1

[[4,4,3,1],[3,3,1]] => [3,3,1] => 110010 => [1,1,3,2] => 1

[[3,2,2,1],[2,1]] => [2,1] => 1010 => [1,2,2] => 1

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Description

The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Map

inner shape

Description

The inner shape of a skew partition.