- FindStat

Identifier

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,-2,3,-4] => 0101 => [1,1,1,1] => 6

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

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

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

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

[-1,2,-3,4] => 1010 => [1,1,1,1] => 6

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

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

[-1,-2,3,-4] => 1101 => [2,1,1] => 5

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

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

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

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

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

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

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

[1,-2,4,-3] => 0101 => [1,1,1,1] => 6

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

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

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

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

[-1,2,-4,3] => 1010 => [1,1,1,1] => 6

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

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

[-1,-2,4,-3] => 1101 => [2,1,1] => 5

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

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

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

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

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

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

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

[1,-3,2,-4] => 0101 => [1,1,1,1] => 6

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

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

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

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

[-1,3,-2,4] => 1010 => [1,1,1,1] => 6

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$F_{1} = 2$

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

$F_{3} = 12 + 12\ q + 12\ q^{2} + 12\ q^{3}$

$F_{4} = 48 + 48\ q + 48\ q^{2} + 96\ q^{3} + 48\ q^{4} + 48\ q^{5} + 48\ q^{6}$

$F_{5} = 240 + 240\ q + 240\ q^{2} + 480\ q^{3} + 480\ q^{4} + 480\ q^{5} + 480\ q^{6} + 480\ q^{7} + 240\ q^{8} + 240\ q^{9} + 240\ q^{10}$

Description

The major index of the composition.
The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .

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

$\pi(i) > 0$ .