- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000885: Binary words ⟶ ℤ

Values

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

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

{{1,2,3}} => [2,3,1] => 10 => 0

{{1,2},{3}} => [2,1,3] => 10 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,4,5},{3}} => [2,4,3,5,1] => 1010 => 0

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

{{1,2,4},{3},{5}} => [2,4,3,1,5] => 1010 => 0

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

{{1,2},{3,4,5}} => [2,1,4,5,3] => 1010 => 0

{{1,2},{3,4},{5}} => [2,1,4,3,5] => 1010 => 0

{{1,2,5},{3},{4}} => [2,5,3,4,1] => 1010 => 0

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

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

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

{{1,3,4,5},{2}} => [3,2,4,5,1] => 1100 => 0

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

{{1,3,4},{2},{5}} => [3,2,4,1,5] => 1100 => 0

{{1,3,5},{2,4}} => [3,4,5,2,1] => 1100 => 0

{{1,3},{2,4,5}} => [3,4,1,5,2] => 1100 => 0

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

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

{{1,3},{2,5},{4}} => [3,5,1,4,2] => 1100 => 0

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

{{1,3},{2},{4},{5}} => [3,2,1,4,5] => 1100 => 0

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

{{1,4},{2,3,5}} => [4,3,5,1,2] => 1010 => 0

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

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

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

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

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

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

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

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

{{1,4,5},{2},{3}} => [4,2,3,5,1] => 1100 => 0

{{1,4},{2,5},{3}} => [4,5,3,1,2] => 1010 => 0

{{1,4},{2},{3,5}} => [4,2,5,1,3] => 1100 => 0

{{1,4},{2},{3},{5}} => [4,2,3,1,5] => 1100 => 0

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

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

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

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

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

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

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

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

{{1,5},{2},{3},{4}} => [5,2,3,4,1] => 1100 => 0

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

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

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

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

{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 10000 => 3

{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 10000 => 3

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 10001 => 3

{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 10001 => 3

{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 10000 => 3

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 10010 => 1

{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 10000 => 3

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 10010 => 1

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 10011 => 3

{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 10010 => 1

{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 10010 => 1

{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 10010 => 1

{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 10011 => 3

{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 10001 => 3

{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 10000 => 3

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 10100 => 1

{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => 10001 => 3

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => 10100 => 1

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 10100 => 1

{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 10100 => 1

{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 10000 => 3

{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 10101 => 1

{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => 10100 => 1

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => 10101 => 1

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => 10100 => 1

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 10110 => 1

{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => 10010 => 1

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Description

The number of critical steps in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation.
The distribution of this statistic on words of length $n$ is
$$ (n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}. $$

Map

to permutation

Description

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

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.