- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

{{1,2}} => [2,1] => [1,2] => 0 => 0

{{1},{2}} => [1,2] => [1,2] => 0 => 0

{{1,2,3}} => [2,3,1] => [1,2,3] => 00 => 0

{{1,2},{3}} => [2,1,3] => [1,2,3] => 00 => 0

{{1,3},{2}} => [3,2,1] => [1,3,2] => 01 => 2

{{1},{2,3}} => [1,3,2] => [1,2,3] => 00 => 0

{{1},{2},{3}} => [1,2,3] => [1,2,3] => 00 => 0

{{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => 000 => 0

{{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => 000 => 0

{{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => 001 => 3

{{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => 000 => 0

{{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => 000 => 0

{{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => 001 => 3

{{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => 010 => 2

{{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => 010 => 2

{{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => 010 => 2

{{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => 000 => 0

{{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => 000 => 0

{{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => 010 => 2

{{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => 001 => 3

{{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => 000 => 0

{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 000 => 0

{{1,2,3,4,5}} => [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0

{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0

{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 4

{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0

{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0

{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 4

{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,2,4,3,5] => 0010 => 3

{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 3

{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 3

{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0

{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0

{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 3

{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 4

{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0

{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0

{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,3,4,5,2] => 0001 => 4

{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,3,4,2,5] => 0010 => 3

{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,3,4,2,5] => 0010 => 3

{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,3,5,2,4] => 0010 => 3

{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,3,2,4,5] => 0100 => 2

{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,3,2,4,5] => 0100 => 2

{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,3,5,2,4] => 0010 => 3

{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,3,2,5,4] => 0101 => 6

{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,3,2,4,5] => 0100 => 2

{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 2

{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,4,5,2,3] => 0010 => 3

{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4,2,3,5] => 0100 => 2

{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,4,2,3,5] => 0100 => 2

{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => 0100 => 2

{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0

{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0

{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,2,3,4] => 0100 => 2

{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 4

{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0

{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0

{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,4,5,2,3] => 0010 => 3

{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,4,2,5,3] => 0101 => 6

{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,2,3,5] => 0100 => 2

{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,4,2,3,5] => 0100 => 2

{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,5,2,4,3] => 0101 => 6

{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 4

{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,2,4,3,5] => 0010 => 3

{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 3

{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,5,2,3,4] => 0100 => 2

{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 3

{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0

{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0

{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,5,2,3,4] => 0100 => 2

{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 3

{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 4

{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0

{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0

{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,2,3,4,6,5] => 00001 => 5

{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,2,3,5,6,4] => 00001 => 5

{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,2,3,5,4,6] => 00010 => 4

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,2,3,5,4,6] => 00010 => 4

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,2,3,6,4,5] => 00010 => 4

{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,2,3,6,4,5] => 00010 => 4

{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,2,3,4,6,5] => 00001 => 5

{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,2,3,4,5,6] => 00000 => 0

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,2,4,5,6,3] => 00001 => 5

{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [1,2,4,5,3,6] => 00010 => 4

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,2,4,5,3,6] => 00010 => 4

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,2,4,6,3,5] => 00010 => 4

{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,2,4,3,5,6] => 00100 => 3

{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,2,4,3,5,6] => 00100 => 3

{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,2,4,6,3,5] => 00010 => 4

{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,2,4,3,6,5] => 00101 => 8

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,2,4,3,5,6] => 00100 => 3

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,2,4,3,5,6] => 00100 => 3

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,2,5,6,3,4] => 00010 => 4

{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [1,2,5,3,4,6] => 00100 => 3

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Description

The sum of the positions of the ones in a binary word.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.