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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001200
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Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,-1] => [2]
=> [1,0,1,0]
=> 2
[-2,1] => [2]
=> [1,0,1,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> 2
[1,-3,2] => [2]
=> [1,0,1,0]
=> 2
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,-1,3] => [2]
=> [1,0,1,0]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,-1] => [2]
=> [1,0,1,0]
=> 2
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> 2
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> 2
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,-3,2,4] => [2]
=> [1,0,1,0]
=> 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-1,3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[-1,3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001491
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(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Values
[2,-1] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,1] => [2]
=> 100 => 010 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[1,3,-2] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,-3,2] => [2]
=> 100 => 010 => 1 = 2 - 1
[-1,3,-2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,-3,2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[2,-1,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,-1,-3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-2,1,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,1,-3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[2,3,-1] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[2,-3,1] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-2,3,1] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-2,-3,-1] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[3,1,-2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[3,-1,2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-3,1,2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-3,-1,-2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[3,2,-1] => [2]
=> 100 => 010 => 1 = 2 - 1
[3,-2,-1] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-3,2,1] => [2]
=> 100 => 010 => 1 = 2 - 1
[-3,-2,1] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> 11110 => 00111 => ? = 3 - 1
[1,2,4,-3] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,2,-4,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[1,-2,-4,3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,2,4,-3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,2,-4,3] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[1,3,-2,4] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[1,-3,2,4] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,3,-2,4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[1,3,4,-2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,3,-4,2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,-3,4,2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,-3,-4,-2] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-1,3,4,-2] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[1,4,2,-3] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,4,-2,3] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,-4,2,3] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[1,-4,-2,-3] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-1,4,2,-3] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[1,4,3,-2] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[1,-4,3,2] => [2]
=> 100 => 010 => 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,4,3,-2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[2,-1,3,4] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,-1,3,-4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[2,-1,-3,4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[-2,1,3,4] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,1,3,-4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-2,1,-3,4] => [2,1]
=> 1010 => 0000 => ? = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> 10110 => 00100 => ? = 3 - 1
[2,1,4,-3] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,1,-4,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,-1,4,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,-1,-4,-3] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,1,4,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,1,-4,-3] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,-1,4,-3] => [2]
=> 100 => 010 => 1 = 2 - 1
[-2,-1,-4,3] => [2]
=> 100 => 010 => 1 = 2 - 1
[2,3,-1,4] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[2,3,-1,-4] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[2,-3,1,4] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[2,-3,1,-4] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-2,3,1,4] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-2,3,1,-4] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[-2,-3,-1,4] => [3]
=> 1000 => 0110 => 2 = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> 10010 => 00010 => ? = 3 - 1
[2,3,4,-1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[2,3,-4,1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[2,-3,4,1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[2,-3,-4,-1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[-2,3,4,1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[-2,3,-4,-1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[-2,-3,4,-1] => [4]
=> 10000 => 01110 => ? = 3 - 1
[2,4,3,-1] => [3]
=> 1000 => 0110 => 2 = 3 - 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: 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: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2 - 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)]
=> ? = 3 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 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)]
=> ? = 3 - 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)]
=> ? = 3 - 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)]
=> ? = 3 - 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)]
=> ? = 3 - 1
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 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
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: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 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] => ? = 3 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 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
[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] => 3 = 2 + 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-2,1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[2,1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,-4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,-1,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,-1,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 3 - 1
[3,2,-1,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,1,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 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
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
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