- FindStat

Your data matches 105 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000944
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The 3-degree of an integer partition. For an integer partition

$\lambda$ , this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by

$\lambda$ . This stupid comment should not be accepted as an edit!

Matching statistic: St001491
(load all 6 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%

Values

[2,-1] => [2]
 => 100 => 001 => 1 = 0 + 1
[-2,1] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,3,-2] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,-3,2] => [2]
 => 100 => 001 => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[2,-1,3] => [2]
 => 100 => 001 => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-2,1,3] => [2]
 => 100 => 001 => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[2,3,-1] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[2,-3,1] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-2,3,1] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[3,1,-2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[3,-1,2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-3,1,2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[3,2,-1] => [2]
 => 100 => 001 => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-3,2,1] => [2]
 => 100 => 001 => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[1,2,4,-3] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,2,-4,3] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,-2,4,-3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[1,-2,-4,3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,2,4,-3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,2,-4,3] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[-1,-2,-4,3] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[1,3,-2,4] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,3,-2,-4] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[1,-3,2,4] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,-3,2,-4] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,3,-2,4] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,3,-2,-4] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[-1,-3,2,4] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,-3,2,-4] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[1,3,4,-2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,3,-4,2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,-3,4,2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,-3,-4,-2] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-1,3,4,-2] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,3,-4,2] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,-3,4,2] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,-3,-4,-2] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[1,4,2,-3] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,4,-2,3] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,-4,2,3] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[1,-4,-2,-3] => [3]
 => 1000 => 0001 => 1 = 0 + 1
[-1,4,2,-3] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,4,-2,3] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,-4,2,3] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-1,-4,-2,-3] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[1,4,3,-2] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,4,-3,-2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[1,-4,3,2] => [2]
 => 100 => 001 => 1 = 0 + 1
[1,-4,-3,2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,4,3,-2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,4,-3,-2] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[-1,-4,3,2] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[-1,-4,-3,2] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[2,-1,3,4] => [2]
 => 100 => 001 => 1 = 0 + 1
[2,-1,3,-4] => [2,1]
 => 1010 => 0011 => 1 = 0 + 1
[2,-1,-3,-4] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[-2,1,-3,-4] => [2,1,1]
 => 10110 => 00111 => ? = 0 + 1
[2,3,-1,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[2,-3,1,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-2,3,1,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-2,-3,-1,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[2,3,4,-1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,3,-4,1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,-3,4,1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,-3,-4,-1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,3,4,1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,3,-4,-1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,-3,4,-1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,-3,-4,1] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,4,1,-3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,4,-1,3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,-4,1,3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,-4,-1,-3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,4,1,3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,4,-1,-3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,-4,1,-3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-2,-4,-1,3] => [4]
 => 10000 => 00001 => ? = 0 + 1
[2,4,-3,-1] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[2,-4,-3,1] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-2,4,-3,1] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-2,-4,-3,-1] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[3,1,-2,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[3,-1,2,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-3,1,2,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[-3,-1,-2,-4] => [3,1]
 => 10010 => 00011 => ? = 0 + 1
[3,1,4,-2] => [4]
 => 10000 => 00001 => ? = 0 + 1
[3,1,-4,2] => [4]
 => 10000 => 00001 => ? = 0 + 1
[3,-1,4,2] => [4]
 => 10000 => 00001 => ? = 0 + 1
[3,-1,-4,-2] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-3,1,4,2] => [4]
 => 10000 => 00001 => ? = 0 + 1
[-3,1,-4,-2] => [4]
 => 10000 => 00001 => ? = 0 + 1

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let

$A_n=K[x]/(x^n)$ . We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.

Matching statistic: St001722
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%

Values

[2,-1] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,2,-1] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,2,4,-3] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,2,-4,3] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,3,-2,4] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-3,2,4] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,-3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-1,-3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,3,4,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,3,-4,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-3,4,2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-3,-4,-2] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[1,4,2,-3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,4,-2,3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-4,2,3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-4,-2,-3] => [3]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[1,4,3,-2] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,4,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-4,3,2] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,-4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,4,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-1,-4,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[2,-1,3,4] => [2]
 => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[2,-1,3,-4] => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[2,3,-1,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[2,-3,1,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-2,3,1,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-2,-3,-1,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[2,3,4,-1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,3,-4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,-3,4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,-3,-4,-1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,3,4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,3,-4,-1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,-3,4,-1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,-3,-4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,4,1,-3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,4,-1,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,-4,1,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,-4,-1,-3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,4,1,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,4,-1,-3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,-4,1,-3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-2,-4,-1,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[2,4,-3,-1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[2,-4,-3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-2,4,-3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-2,-4,-3,-1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[3,1,-2,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[3,-1,2,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-3,1,2,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[-3,-1,-2,-4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[3,1,4,-2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[3,1,-4,2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[3,-1,4,2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[3,-1,-4,-2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-3,1,4,2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1
[-3,1,-4,-2] => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 0 + 1

Description

The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence

$01$ , or a trailing

$0$ . A peak is a subsequence

$10$ or a trailing

$1$ . Let

$P$ be the lattice on binary words of length

$n$ , where the covering elements of a word are obtained by replacing a valley with a peak. An interval

$[w_1, w_2]$ in

$P$ is small if

$w_2$ is obtained from

$w_1$ by replacing some valleys with peaks. This statistic counts the number of chains

$w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word

$0110$ :

$0110 < 1011 < 1101 < 1110 < 1111$ and

$0110 < 1010 < 1101 < 1110 < 1111.$

Matching statistic: St000782
(load all 4 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%

Values

[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-2,1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[2,1,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,1,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,-1,4,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[2,-1,-4,-3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,1,4,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-2,1,-4,-3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,-1,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[-2,-1,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[2,3,4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[2,3,-4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[2,-3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[2,-3,-4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[-2,3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[-2,3,-4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[3,2,-1,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-3,2,1,4] => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1

Description

The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin

$([1,3],[2,4])$ , insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2])

$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$

Matching statistic: St001583

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%

Values

[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[-2,1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[2,1,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,1,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,-1,4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 0 + 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 0 + 3
[2,-1,-4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,1,4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 0 + 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 0 + 3
[-2,1,-4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,-1,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[-2,-1,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[2,3,4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[2,3,-4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[2,-3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[2,-3,-4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[-2,3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[-2,3,-4,-1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[3,2,-1,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[-3,2,1,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3

Description

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.

Matching statistic: St001771
(load all 16 compositions to match this statistic)

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs

$1\leq i < j\leq n$ such that

$0 < \pi(i) < -\pi(j)$ .

Matching statistic: St001870
(load all 16 compositions to match this statistic)

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , this is the number of positive entries followed by a negative entry in

$\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$ .

Matching statistic: St001895
(load all 15 compositions to match this statistic)

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%

Values

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

Description

The oddness of a signed permutation. The direct sum of two signed permutations

$\sigma\in\mathfrak H_k$ and

$\tau\in\mathfrak H_m$ is the signed permutation in

$\mathfrak H_{k+m}$ obtained by concatenating

$\sigma$ with the result of increasing the absolute value of every entry in

$\tau$ by

$k$ . This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.

Matching statistic: St001851
(load all 11 compositions to match this statistic)

Mapping

St001851: Signed permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of Hecke atoms of a signed permutation. For a signed permutation

$z\in\mathfrak H_n$ , this is the cardinality of the set

$\{ w\in\mathfrak H_n | w^{-1} \star w = z\},$ where

$\star$ denotes the Demazure product. Note that

$w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.

Matching statistic: St001772

Mapping

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of occurrences of the signed pattern 12 in a signed permutation. This is the number of pairs

$1\leq i < j\leq n$ such that

$0 < \pi(i) < \pi(j)$ .

The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000068The number of minimal elements in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001490The number of connected components of a skew partition. St001768The number of reduced words of a signed permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000911The number of maximal antichains of maximal size in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000635The number of strictly order preserving maps of a poset into itself. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree.