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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000148
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,-3] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 1
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 1
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [3,1]
=> [1]
=> 1
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 1
[-1,3,-2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-4,3,2] => [2]
=> [1,1]
=> [1]
=> 1
[-1,4,-3,-2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-4,-3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
Description
The number of odd parts of a partition.
Matching statistic: St000475
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,-3] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 1
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 1
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [3,1]
=> [1]
=> 1
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 1
[-1,3,-2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-4,3,2] => [2]
=> [1,1]
=> [1]
=> 1
[-1,4,-3,-2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-4,-3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001247
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,-3] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 1
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 1
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [3,1]
=> [1]
=> 1
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 1
[-1,3,-2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-4,3,2] => [2]
=> [1,1]
=> [1]
=> 1
[-1,4,-3,-2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-4,-3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,-3] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2] => [2]
=> [1,1]
=> [1]
=> 1
[2,-1,3] => [2]
=> [1,1]
=> [1]
=> 1
[-2,1,3] => [2]
=> [1,1]
=> [1]
=> 1
[2,3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[2,-3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-2,-3,-1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,-1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-3,-1,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,-1] => [2]
=> [1,1]
=> [1]
=> 1
[-3,2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [3,1]
=> [1]
=> 1
[1,2,4,-3] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,-4,3] => [2]
=> [1,1]
=> [1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-2,-4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,-2,4] => [2]
=> [1,1]
=> [1]
=> 1
[1,-3,2,4] => [2]
=> [1,1]
=> [1]
=> 1
[-1,3,-2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-3,2,-4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,-4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,4,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,-2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,-2] => [2]
=> [1,1]
=> [1]
=> 1
[1,-4,3,2] => [2]
=> [1,1]
=> [1]
=> 1
[-1,4,-3,-2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[-1,-4,-3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
Description
Sum of the odd parts of a partition.
Matching statistic: St000745
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,-2,-3] => [1,1,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,3,-2] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-3,2] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-2,1,3] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,3,-1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[2,-3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-2,3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-2,-3,-1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,1,-2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,-1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-3,1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-3,-1,-2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,2,-1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-3,2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,-3,-4] => [1,1,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-1,2,-3,-4] => [1,1,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-1,-2,3,-4] => [1,1,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,2,4,-3] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,-2,4,-3] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[-1,-2,-4,3] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[1,3,-2,4] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-3,2,4] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,3,-2,-4] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[-1,-3,2,-4] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[1,3,4,-2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,-4,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,-3,4,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,2,-3] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,4,-2,3] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,-4,2,3] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,3,-2] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-4,3,2] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,4,-3,-2] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
[-1,-4,-3,2] => [2,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[2,-1] => [2]
=> [1,1]
=> 110 => 1
[-2,1] => [2]
=> [1,1]
=> 110 => 1
[-1,-2,-3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,3,-2] => [2]
=> [1,1]
=> 110 => 1
[1,-3,2] => [2]
=> [1,1]
=> 110 => 1
[2,-1,3] => [2]
=> [1,1]
=> 110 => 1
[-2,1,3] => [2]
=> [1,1]
=> 110 => 1
[2,3,-1] => [3]
=> [1,1,1]
=> 1110 => 2
[2,-3,1] => [3]
=> [1,1,1]
=> 1110 => 2
[-2,3,1] => [3]
=> [1,1,1]
=> 1110 => 2
[-2,-3,-1] => [3]
=> [1,1,1]
=> 1110 => 2
[3,1,-2] => [3]
=> [1,1,1]
=> 1110 => 2
[3,-1,2] => [3]
=> [1,1,1]
=> 1110 => 2
[-3,1,2] => [3]
=> [1,1,1]
=> 1110 => 2
[-3,-1,-2] => [3]
=> [1,1,1]
=> 1110 => 2
[3,2,-1] => [2]
=> [1,1]
=> 110 => 1
[-3,2,1] => [2]
=> [1,1]
=> 110 => 1
[1,-2,-3,-4] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,2,-3,-4] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,-2,3,-4] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,-2,-3,4] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,2,4,-3] => [2]
=> [1,1]
=> 110 => 1
[1,2,-4,3] => [2]
=> [1,1]
=> 110 => 1
[-1,-2,4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[-1,-2,-4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[1,3,-2,4] => [2]
=> [1,1]
=> 110 => 1
[1,-3,2,4] => [2]
=> [1,1]
=> 110 => 1
[-1,3,-2,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[-1,-3,2,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[1,3,4,-2] => [3]
=> [1,1,1]
=> 1110 => 2
[1,3,-4,2] => [3]
=> [1,1,1]
=> 1110 => 2
[1,-3,4,2] => [3]
=> [1,1,1]
=> 1110 => 2
[1,-3,-4,-2] => [3]
=> [1,1,1]
=> 1110 => 2
[-1,3,4,-2] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,3,-4,2] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,-3,4,2] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,-3,-4,-2] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,4,2,-3] => [3]
=> [1,1,1]
=> 1110 => 2
[1,4,-2,3] => [3]
=> [1,1,1]
=> 1110 => 2
[1,-4,2,3] => [3]
=> [1,1,1]
=> 1110 => 2
[1,-4,-2,-3] => [3]
=> [1,1,1]
=> 1110 => 2
[-1,4,2,-3] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,4,-2,3] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,-4,2,3] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-1,-4,-2,-3] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[1,4,3,-2] => [2]
=> [1,1]
=> 110 => 1
[1,-4,3,2] => [2]
=> [1,1]
=> 110 => 1
[-1,4,-3,-2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[-1,-4,-3,2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[2,-1,3,4] => [2]
=> [1,1]
=> 110 => 1
[2,-1,-3,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[-2,1,3,4] => [2]
=> [1,1]
=> 110 => 1
[-2,1,-3,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 0
[2,1,4,-3] => [2]
=> [1,1]
=> 110 => 1
[2,1,-4,3] => [2]
=> [1,1]
=> 110 => 1
[2,-1,4,3] => [2]
=> [1,1]
=> 110 => 1
[2,-1,-4,-3] => [2]
=> [1,1]
=> 110 => 1
[-2,1,4,3] => [2]
=> [1,1]
=> 110 => 1
[-2,1,-4,-3] => [2]
=> [1,1]
=> 110 => 1
[-2,-1,4,-3] => [2]
=> [1,1]
=> 110 => 1
[-2,-1,-4,3] => [2]
=> [1,1]
=> 110 => 1
[2,3,-1,4] => [3]
=> [1,1,1]
=> 1110 => 2
[2,3,-1,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[2,-3,1,4] => [3]
=> [1,1,1]
=> 1110 => 2
[2,-3,1,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-2,3,1,4] => [3]
=> [1,1,1]
=> 1110 => 2
[-2,3,1,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-2,-3,-1,4] => [3]
=> [1,1,1]
=> 1110 => 2
[-2,-3,-1,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[2,3,4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,3,-4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-3,-4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,3,-4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-3,4,-1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-3,-4,1] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,-1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,-4,-1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,4,-1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-4,1,-3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-2,-4,-1,3] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[2,4,3,-1] => [3]
=> [1,1,1]
=> 1110 => 2
[2,4,-3,-1] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[2,-4,-3,1] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-2,4,-3,1] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-2,-4,-3,-1] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[3,1,-2,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[3,-1,2,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-3,1,2,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[-3,-1,-2,-4] => [3,1]
=> [2,1,1]
=> 10110 => ? = 2
[3,1,4,-2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,1,-4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,-1,4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[3,-1,-4,-2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
[-3,1,4,2] => [4]
=> [1,1,1,1]
=> 11110 => ? = 3
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: St000782
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: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[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)]
=> ? = 3
[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)]
=> ? = 3
[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)]
=> ? = 3
[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)]
=> ? = 3
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,-4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,-4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,-4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,-4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,2,3,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-4,2,3,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,-3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-4,3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-4,3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-4,-3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-4,-3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,3,5,-4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,3,-5,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,4,-3,5] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,-4,3,5] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,5,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,-5,4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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: St001722
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,-4,1,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,-4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,4,1,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,-4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-3,-4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,2,3,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-4,2,3,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,-3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-4,3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-4,3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-4,-3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[-4,-3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,3,5,-4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,3,-5,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,4,-3,5] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,-4,3,5] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,5,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,2,-5,4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 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: St001816
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 3
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[3,4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[3,4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[3,-4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[3,-4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,-4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-3,-4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,2,3,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-4,2,3,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,-3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-4,3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-4,3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-4,-3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[-4,-3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,3,5,-4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,3,-5,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,4,-3,5] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,-4,3,5] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,5,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,2,-5,4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001207
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 1
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 3 + 1
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[3,4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[3,4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[3,-4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[3,-4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,4,1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,4,-1,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,-4,1,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-3,-4,-1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,2,3,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-4,2,3,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[4,-3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-4,3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-4,3,-2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-4,-3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[-4,-3,-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,3,5,-4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,3,-5,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,4,-3,5] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,-4,3,5] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,5,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,2,-5,4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000075The orbit size of a standard tableau under promotion. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000529The number of permutations whose descent word is the given binary word.
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