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Matching statistic: St000255
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Mapping
St000255: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 3
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 6
[1,2,5,3,4] => 6
[1,2,5,4,3] => 14
[1,3,2,4,5] => 2
[1,3,2,5,4] => 8
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 8
[1,3,5,4,2] => 11
[1,4,2,3,5] => 3
[1,4,2,5,3] => 8
[1,4,3,2,5] => 5
[1,4,3,5,2] => 7
[1,4,5,2,3] => 6
Description
The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Matching statistic: St001770
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001770: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 6%
Values
[1] => [1] => [1] => [-1] => 1
[1,2] => [2,1] => [2,1] => [-2,-1] => 1
[2,1] => [1,2] => [1,2] => [-1,-2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 1
[1,3,2] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 2
[2,1,3] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 1
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 3
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 2
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 3
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 3
[1,4,3,2] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 5
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 3
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 1
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 2
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 2
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 1
[4,1,3,2] => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 2
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => [-4,-5,-3,-2,-1] => ? = 4
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => [-5,-3,-4,-2,-1] => ? = 3
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => [-3,-5,-4,-2,-1] => ? = 6
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => [-4,-3,-5,-2,-1] => ? = 6
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 14
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [-5,-4,-2,-3,-1] => ? = 2
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => [-4,-5,-2,-3,-1] => ? = 8
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => ? = 3
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 4
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => ? = 8
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => ? = 11
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => ? = 3
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => ? = 8
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => ? = 5
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 7
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 6
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 9
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => ? = 4
[1,5,2,4,3] => [3,4,2,5,1] => [3,4,2,5,1] => [-3,-4,-2,-5,-1] => ? = 11
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => ? = 7
[1,5,3,4,2] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 10
[1,5,4,2,3] => [3,2,4,5,1] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => ? = 9
[1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 14
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => [-5,-4,-3,-1,-2] => ? = 1
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 4
[2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => [-5,-3,-4,-1,-2] => ? = 3
[2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 6
[2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 6
[2,1,5,4,3] => [3,4,5,1,2] => [3,4,5,1,2] => [-3,-4,-5,-1,-2] => ? = 14
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => [-5,-4,-1,-3,-2] => ? = 1
[2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => ? = 4
[2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => [-5,-1,-4,-3,-2] => ? = 1
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 1
[2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => [-4,-1,-5,-3,-2] => ? = 3
[2,3,5,4,1] => [1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 3
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => [-5,-3,-1,-4,-2] => ? = 2
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => ? = 5
[2,4,3,1,5] => [5,1,3,4,2] => [5,1,3,4,2] => [-5,-1,-3,-4,-2] => ? = 2
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 2
[2,4,5,1,3] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 3
[2,4,5,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 3
[2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => [-4,-3,-1,-5,-2] => ? = 3
[2,5,1,4,3] => [3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => ? = 8
[2,5,3,1,4] => [4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => ? = 3
[2,5,3,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 3
[2,5,4,1,3] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 5
[2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 5
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => [-5,-4,-2,-1,-3] => ? = 1
[3,1,2,5,4] => [4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => ? = 4
[3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => ? = 2
[3,1,4,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 3
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 5
[3,1,5,4,2] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 8
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => ? = 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 3
[3,4,2,5,1] => [1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => 1
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 2
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 2
[4,2,3,5,1] => [1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 2
[4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => 1
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 1
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 1
Description
The number of facets of a certain subword complex associated with the signed permutation. Let $Q=[1,\dots,n,1,\dots,n,\dots,1,\dots,n]$ be the word of length $n^2$, and let $\pi$ be a signed permutation. Then this statistic yields the number of facets of the subword complex $\Delta(Q, \pi)$.