- FindStat

Identifier

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤ

Values

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

[1,2] => {{1},{2}} => [1,1] => 11 => 0

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

[1,2,3] => {{1},{2},{3}} => [1,1,1] => 111 => 0

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

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

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

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

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

[1,2,3,4] => {{1},{2},{3},{4}} => [1,1,1,1] => 1111 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,1,1,1,1] => 11111 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,3,4,5] => {{1,2},{3},{4},{5}} => [2,1,1,1] => 10111 => 1

[2,1,3,5,4] => {{1,2},{3},{4,5}} => [2,1,2] => 10110 => 2

[2,1,4,3,5] => {{1,2},{3,4},{5}} => [2,2,1] => 10101 => 2

[2,1,4,5,3] => {{1,2},{3,4,5}} => [2,3] => 10100 => 2

[2,1,5,3,4] => {{1,2},{3,5},{4}} => [2,2,1] => 10101 => 2

[2,1,5,4,3] => {{1,2},{3,5},{4}} => [2,2,1] => 10101 => 2

[2,3,1,4,5] => {{1,2,3},{4},{5}} => [3,1,1] => 10011 => 1

[2,3,1,5,4] => {{1,2,3},{4,5}} => [3,2] => 10010 => 2

[2,3,4,1,5] => {{1,2,3,4},{5}} => [4,1] => 10001 => 1

[2,3,4,5,1] => {{1,2,3,4,5}} => [5] => 10000 => 1

[2,3,5,1,4] => {{1,2,3,5},{4}} => [4,1] => 10001 => 1

[2,3,5,4,1] => {{1,2,3,5},{4}} => [4,1] => 10001 => 1

[2,4,1,3,5] => {{1,2,4},{3},{5}} => [3,1,1] => 10011 => 1

[2,4,1,5,3] => {{1,2,4,5},{3}} => [4,1] => 10001 => 1

[2,4,3,1,5] => {{1,2,4},{3},{5}} => [3,1,1] => 10011 => 1

[2,4,3,5,1] => {{1,2,4,5},{3}} => [4,1] => 10001 => 1

[2,4,5,1,3] => {{1,2,4},{3,5}} => [3,2] => 10010 => 2

[2,4,5,3,1] => {{1,2,4},{3,5}} => [3,2] => 10010 => 2

[2,5,1,3,4] => {{1,2,5},{3},{4}} => [3,1,1] => 10011 => 1

[2,5,1,4,3] => {{1,2,5},{3},{4}} => [3,1,1] => 10011 => 1

[2,5,3,1,4] => {{1,2,5},{3},{4}} => [3,1,1] => 10011 => 1

[2,5,3,4,1] => {{1,2,5},{3},{4}} => [3,1,1] => 10011 => 1

[2,5,4,1,3] => {{1,2,5},{3,4}} => [3,2] => 10010 => 2

[2,5,4,3,1] => {{1,2,5},{3,4}} => [3,2] => 10010 => 2

[3,1,2,4,5] => {{1,3},{2},{4},{5}} => [2,1,1,1] => 10111 => 1

[3,1,2,5,4] => {{1,3},{2},{4,5}} => [2,1,2] => 10110 => 2

[3,1,4,2,5] => {{1,3,4},{2},{5}} => [3,1,1] => 10011 => 1

[3,1,4,5,2] => {{1,3,4,5},{2}} => [4,1] => 10001 => 1

[3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,1,1] => 10011 => 1

[3,1,5,4,2] => {{1,3,5},{2},{4}} => [3,1,1] => 10011 => 1

[3,2,1,4,5] => {{1,3},{2},{4},{5}} => [2,1,1,1] => 10111 => 1

[3,2,1,5,4] => {{1,3},{2},{4,5}} => [2,1,2] => 10110 => 2

[3,2,4,1,5] => {{1,3,4},{2},{5}} => [3,1,1] => 10011 => 1

[3,2,4,5,1] => {{1,3,4,5},{2}} => [4,1] => 10001 => 1

[3,2,5,1,4] => {{1,3,5},{2},{4}} => [3,1,1] => 10011 => 1

[3,2,5,4,1] => {{1,3,5},{2},{4}} => [3,1,1] => 10011 => 1

[3,4,1,2,5] => {{1,3},{2,4},{5}} => [2,2,1] => 10101 => 2

[3,4,1,5,2] => {{1,3},{2,4,5}} => [2,3] => 10100 => 2

[3,4,2,1,5] => {{1,3},{2,4},{5}} => [2,2,1] => 10101 => 2

[3,4,2,5,1] => {{1,3},{2,4,5}} => [2,3] => 10100 => 2

[3,4,5,1,2] => {{1,3,5},{2,4}} => [3,2] => 10010 => 2

[3,4,5,2,1] => {{1,3,5},{2,4}} => [3,2] => 10010 => 2

[3,5,1,2,4] => {{1,3},{2,5},{4}} => [2,2,1] => 10101 => 2

[3,5,1,4,2] => {{1,3},{2,5},{4}} => [2,2,1] => 10101 => 2

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Description

The number of descents of a binary word.

Map

to composition

Description

The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.