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Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: 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] => [3,1,2] => 10 => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [5,6,1,2,3,4] => 01000 => 1

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

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

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [4,6,1,2,3,5] => 01000 => 1

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

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [4,5,1,2,3,6] => 01000 => 1

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

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

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

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

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

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

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

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [3,6,1,2,4,5] => 01000 => 1

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

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [3,5,1,2,4,6] => 01000 => 1

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

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

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

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

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

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [3,4,1,2,6,5] => 01001 => 2

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,4,1,2,5,6] => 01000 => 1

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 3\ q^{2}$

$F_{5} = 1 + 26\ q + 25\ q^{2}$

$F_{6} = 1 + 57\ q + 130\ q^{2} + 15\ q^{3}$

$F_{7} = 1 + 120\ q + 546\ q^{2} + 210\ q^{3}$

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

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.