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Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001811: Permutations ⟶ ℤ

Values

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

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$F_{1} = 2$

$F_{2} = 3 + q$

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

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

Description

The Castelnuovo-Mumford regularity of a permutation.
The Castelnuovo-Mumford regularity of a permutation

$\sigma$ is the Castelnuovo-Mumford regularity of the matrix Schubert variety

$X_\sigma$ .
Equivalently, it is the difference between the degrees of the Grothendieck polynomial and the Schubert polynomial for

$\sigma$ . It can be computed by subtracting the Coxeter length St000018The number of inversions of a permutation. from the Rajchgot index St001759The Rajchgot index of a permutation..

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

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.