- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,0] => [1,2,4,3] => 001 => 0

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

[1,0,1,1,0,1,0,0] => [1,3,4,2] => 001 => 0

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

[1,1,0,0,1,0,1,0] => [2,1,3,4] => 100 => 1

[1,1,0,0,1,1,0,0] => [2,1,4,3] => 101 => 1

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

[1,1,0,1,0,1,0,0] => [2,3,4,1] => 001 => 0

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

[1,1,1,0,0,0,1,0] => [3,1,2,4] => 100 => 1

[1,1,1,0,0,1,0,0] => [3,1,4,2] => 101 => 1

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

[1,1,1,1,0,0,0,0] => [4,1,2,3] => 100 => 1

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0000 => 0

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0001 => 0

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

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 0001 => 0

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

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

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

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

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 0001 => 0

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1000 => 1

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1001 => 1

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

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 1001 => 1

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

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

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

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

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0001 => 0

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

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

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

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

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

[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 1000 => 1

[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 1001 => 1

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

[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 1001 => 1

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

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

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

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

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

[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 1000 => 1

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 1001 => 1

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

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

[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1000 => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 00000 => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 00001 => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 00010 => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 00001 => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 00010 => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 00100 => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 00101 => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 00010 => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 00001 => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 00010 => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 00100 => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 00101 => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 00010 => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 00100 => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 01000 => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 01001 => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 01010 => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 01001 => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 01010 => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 00100 => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 00101 => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 00010 => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 00001 => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 00010 => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 00100 => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 00101 => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 00010 => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 00100 => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 01000 => 0

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 01001 => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 01010 => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 01001 => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 01010 => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 00100 => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 00101 => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 00010 => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 00100 => 0

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 01000 => 0

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Description

The number of leading ones in a binary word.

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.

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.