- FindStat

Identifier

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[3,-2,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

[-3,-2,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

[1,2,-3,-4] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2

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

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

[1,-2,-3,4] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

[1,-2,-3,-4] => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3

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

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

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

[-1,2,-3,-4] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 3

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

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

[-1,-2,-3,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3

[-1,-2,-3,-4] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4

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

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

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

[1,2,-4,-3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2

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

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

[1,-2,-4,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

[1,-2,-4,-3] => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3

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

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

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

[-1,2,-4,-3] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 3

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

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

[-1,-2,-4,3] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3

[-1,-2,-4,-3] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4

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

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

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

[1,3,-2,-4] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2

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

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

[1,-3,-2,4] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

[1,-3,-2,-4] => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3

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

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

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

>>> Load all 4282 entries. <<<

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$F_{1} = 1 + q$

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

$F_{3} = 6 + 18\ q + 18\ q^{2} + 6\ q^{3}$

$F_{4} = 24 + 96\ q + 144\ q^{2} + 96\ q^{3} + 24\ q^{4}$

$F_{5} = 120 + 600\ q + 1320\ q^{2} + 1080\ q^{3} + 600\ q^{4} + 120\ q^{5}$

Description

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

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

bounce path

Description

The bounce path determined by an integer composition.

Map

signs

Description

The binary word recording the signs of a signed permutation.
This map sends a signed permutation

$\pi\in\mathfrak H_n$ to the binary word

$w$ of length

$n$ such that

$w_i = 0$ if

$\pi(i) > 0$ .