- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤ

Values

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

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

{{1,2,3}} => [2,3,1] => [3,2,1] => 11 => 2

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

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

{{1},{2,3}} => [1,3,2] => [1,3,2] => 01 => 0

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

{{1,2,3,4}} => [2,3,4,1] => [4,3,2,1] => 111 => 3

{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => 110 => 2

{{1,2,4},{3}} => [2,4,3,1] => [3,4,2,1] => 110 => 2

{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 101 => 1

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

{{1,3,4},{2}} => [3,2,4,1] => [2,4,3,1] => 101 => 1

{{1,3},{2,4}} => [3,4,1,2] => [4,1,3,2] => 110 => 2

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

{{1,4},{2,3}} => [4,3,2,1] => [3,2,4,1] => 110 => 2

{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => 011 => 0

{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 010 => 0

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

{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => 010 => 0

{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 001 => 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] => [5,4,3,2,1] => 1111 => 4

{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 3

{{1,2,3,5},{4}} => [2,3,5,4,1] => [4,5,3,2,1] => 1110 => 3

{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 2

{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 2

{{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,4,2,1] => 1101 => 2

{{1,2,4},{3,5}} => [2,4,5,1,3] => [5,2,1,4,3] => 1110 => 3

{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,4,2,1,5] => 1100 => 2

{{1,2,5},{3,4}} => [2,5,4,3,1] => [4,3,5,2,1] => 1110 => 3

{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1

{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1

{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,2,1] => 1100 => 2

{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => 1010 => 1

{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1

{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1

{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,4,3,1] => 1011 => 1

{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,1,5,3,2] => 1110 => 3

{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,4,3,1,5] => 1010 => 1

{{1,3,5},{2,4}} => [3,4,5,2,1] => [5,2,4,3,1] => 1110 => 3

{{1,3},{2,4,5}} => [3,4,1,5,2] => [5,4,1,3,2] => 1101 => 2

{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,1,3,2,5] => 1100 => 2

{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,3,1] => 1010 => 1

{{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,5,1,3,2] => 1100 => 2

{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => 1001 => 1

{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => 1000 => 1

{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,2,5,4,1] => 1101 => 2

{{1,4},{2,3,5}} => [4,3,5,1,2] => [5,3,1,4,2] => 1110 => 3

{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,2,4,1,5] => 1100 => 2

{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,3,2,5,1] => 1110 => 3

{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 0

{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => 0110 => 0

{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,2,4,5,1] => 1100 => 2

{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,3,2] => 0110 => 0

{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 0

{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0

{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,3,5,4,1] => 1001 => 1

{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,5,1,4,2] => 1100 => 2

{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,5,1,4,3] => 1010 => 1

{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,3,4,1,5] => 1000 => 1

{{1,5},{2,4},{3}} => [5,4,3,2,1] => [3,4,2,5,1] => 1100 => 2

{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,4,2] => 0101 => 0

{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,2,4,3] => 0110 => 0

{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => 0100 => 0

{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,4,3,5,1] => 1010 => 1

{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,4,3,5,2] => 0110 => 0

{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => 0011 => 0

{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0

{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,3,4,5,1] => 1000 => 1

{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,4,5,2] => 0100 => 0

{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => 0010 => 0

{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 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] => [6,5,4,3,2,1] => 11111 => 5

{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,4,3,2,1,6] => 11110 => 4

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [5,6,4,3,2,1] => 11110 => 4

{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [4,3,2,1,6,5] => 11101 => 3

{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,3,2,1,5,6] => 11100 => 3

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [4,6,5,3,2,1] => 11101 => 3

{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [6,3,2,1,5,4] => 11110 => 4

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [4,5,3,2,1,6] => 11100 => 3

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [5,4,6,3,2,1] => 11110 => 4

{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 11011 => 2

{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 11010 => 2

{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [4,5,6,3,2,1] => 11100 => 3

{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,5,6,4] => 11010 => 2

{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 11001 => 2

{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 11000 => 2

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [3,6,5,4,2,1] => 11011 => 2

{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [5,2,1,6,4,3] => 11110 => 4

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [3,5,4,2,1,6] => 11010 => 2

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,3,5,4,2,1] => 11110 => 4

{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [6,5,2,1,4,3] => 11101 => 3

{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [5,2,1,4,3,6] => 11100 => 3

{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [3,5,6,4,2,1] => 11010 => 2

{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [5,6,2,1,4,3] => 11100 => 3

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [3,4,2,1,6,5] => 11001 => 2

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,4,2,1,5,6] => 11000 => 2

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [4,3,6,5,2,1] => 11101 => 3

{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [6,4,2,1,5,3] => 11110 => 4

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Description

The number of leading ones in a binary word.

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.

Map

to permutation

Description

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

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].