- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

[1,0] => [1] => [1] => 1 => 1

[1,0,1,0] => [1,2] => [2] => 10 => 1

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

[1,0,1,0,1,0] => [1,2,3] => [3] => 100 => 1

[1,0,1,1,0,0] => [1,3,2] => [2,1] => 101 => 2

[1,1,0,0,1,0] => [2,1,3] => [1,2] => 110 => 2

[1,1,0,1,0,0] => [2,3,1] => [2,1] => 101 => 2

[1,1,1,0,0,0] => [3,1,2] => [1,2] => 110 => 2

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [4] => 1000 => 1

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [3,1] => 1001 => 2

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [2,2] => 1010 => 2

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [3,1] => 1001 => 2

[1,0,1,1,1,0,0,0] => [1,4,2,3] => [2,2] => 1010 => 2

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,3] => 1100 => 2

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,1] => 1101 => 3

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,2] => 1010 => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [3,1] => 1001 => 2

[1,1,0,1,1,0,0,0] => [2,4,1,3] => [2,2] => 1010 => 2

[1,1,1,0,0,0,1,0] => [3,1,2,4] => [1,3] => 1100 => 2

[1,1,1,0,0,1,0,0] => [3,1,4,2] => [1,2,1] => 1101 => 3

[1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,2] => 1010 => 2

[1,1,1,1,0,0,0,0] => [4,1,2,3] => [1,3] => 1100 => 2

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [5] => 10000 => 1

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [4,1] => 10001 => 2

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [3,2] => 10010 => 2

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [4,1] => 10001 => 2

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [3,2] => 10010 => 2

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [2,3] => 10100 => 2

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [2,2,1] => 10101 => 3

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [3,2] => 10010 => 2

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [4,1] => 10001 => 2

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [3,2] => 10010 => 2

[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [2,3] => 10100 => 2

[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [2,2,1] => 10101 => 3

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [3,2] => 10010 => 2

[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [2,3] => 10100 => 2

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,4] => 11000 => 2

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,3,1] => 11001 => 3

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,2] => 11010 => 3

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,3,1] => 11001 => 3

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [1,2,2] => 11010 => 3

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3] => 10100 => 2

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,2,1] => 10101 => 3

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [3,2] => 10010 => 2

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [4,1] => 10001 => 2

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [3,2] => 10010 => 2

[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [2,3] => 10100 => 2

[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [2,2,1] => 10101 => 3

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [3,2] => 10010 => 2

[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [2,3] => 10100 => 2

[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [1,4] => 11000 => 2

[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [1,3,1] => 11001 => 3

[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [1,2,2] => 11010 => 3

[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [1,3,1] => 11001 => 3

[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [1,2,2] => 11010 => 3

[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [2,3] => 10100 => 2

[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [2,2,1] => 10101 => 3

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [3,2] => 10010 => 2

[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [2,3] => 10100 => 2

[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [1,4] => 11000 => 2

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [1,3,1] => 11001 => 3

[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [1,2,2] => 11010 => 3

[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [2,3] => 10100 => 2

[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [1,4] => 11000 => 2

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [6] => 100000 => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [5,1] => 100001 => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [4,2] => 100010 => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [5,1] => 100001 => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [4,2] => 100010 => 2

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [3,3] => 100100 => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [3,2,1] => 100101 => 3

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [4,2] => 100010 => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [5,1] => 100001 => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [4,2] => 100010 => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [3,3] => 100100 => 2

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [3,2,1] => 100101 => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [4,2] => 100010 => 2

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [3,3] => 100100 => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [2,4] => 101000 => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [2,3,1] => 101001 => 3

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [2,2,2] => 101010 => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [2,3,1] => 101001 => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [2,2,2] => 101010 => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [3,3] => 100100 => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [3,2,1] => 100101 => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [4,2] => 100010 => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [5,1] => 100001 => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [4,2] => 100010 => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [3,3] => 100100 => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [3,2,1] => 100101 => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [4,2] => 100010 => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [3,3] => 100100 => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [2,4] => 101000 => 2

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [2,3,1] => 101001 => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [2,2,2] => 101010 => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [2,3,1] => 101001 => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [2,2,2] => 101010 => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [3,3] => 100100 => 2

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [3,2,1] => 100101 => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [4,2] => 100010 => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [3,3] => 100100 => 2

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

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

$F_{5} = q + 26\ q^{2} + 15\ q^{3}$

$F_{6} = q + 57\ q^{2} + 69\ q^{3} + 5\ q^{4}$

$F_{7} = q + 120\ q^{2} + 252\ q^{3} + 56\ q^{4}$

Description

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

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.

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

$0$ .