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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000255
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(load all 11 compositions to match this statistic)
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
[2,3,5,4,1] => 3
[2,4,3,5,1] => 2
[2,4,5,3,1] => 3
[2,5,3,4,1] => 3
[2,5,4,3,1] => 5
[3,2,4,5,1] => 1
[3,2,5,4,1] => 3
[3,4,2,5,1] => 1
[3,4,5,2,1] => 1
[3,5,2,4,1] => 2
[3,5,4,2,1] => 2
[4,2,3,5,1] => 1
[4,2,5,3,1] => 2
[4,3,2,5,1] => 1
[4,3,5,2,1] => 1
[4,5,2,3,1] => 1
[4,5,3,2,1] => 1
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
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001770: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001770: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[2,3,5,4,1] => [1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 3
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 2
[2,4,5,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 3
[2,5,3,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 3
[2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 5
[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 $\Delta(Q, \pi)$.
Matching statistic: St000454
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2] => ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2 - 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 5 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 1 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 5 - 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 - 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0 = 1 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001232
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 1
[1,2] => [1,2] => [2]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [1,2] => [2]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[2,3,1] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,1] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 5
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,3,4,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[3,4,1,2] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,4,2,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[4,2,3,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,2,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,3,5,4,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,4,3,5,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[2,4,5,3,1] => [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,5,4,3,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 5
[3,2,4,5,1] => [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[3,2,5,4,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[3,4,2,5,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[3,4,5,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,5,2,4,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[3,5,4,2,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[4,2,3,5,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[4,2,5,3,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2,5,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[4,3,5,2,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[4,5,2,3,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,2,4,3,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[5,3,2,4,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[5,3,4,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001878
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 3
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 5
[2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 2
[3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[3,1,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 3
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 5
[3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3
[3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2
[3,5,4,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 2
[4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[4,2,5,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 2 - 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 3 - 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 5 - 1
[2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 2 - 1
[3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[3,1,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2 - 1
[3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 3 - 1
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 5 - 1
[3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[3,5,4,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 2 - 1
[4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[4,2,5,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1 - 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 2 - 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 3 - 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 5 - 1
[2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 2 - 1
[3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[3,1,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2 - 1
[3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 3 - 1
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 5 - 1
[3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 1
[3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[3,5,4,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 2 - 1
[4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[4,2,5,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 2 - 1
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1 - 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001857
(load all 67 compositions to match this statistic)
(load all 67 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [2,1] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[2,1,3] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[3,2,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => ? = 1 - 1
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => ? = 3 - 1
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => ? = 2 - 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => ? = 3 - 1
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => ? = 3 - 1
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => ? = 5 - 1
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => ? = 1 - 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => ? = 3 - 1
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => ? = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ? = 1 - 1
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => ? = 2 - 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => ? = 2 - 1
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => ? = 1 - 1
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => ? = 2 - 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => ? = 1 - 1
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => ? = 1 - 1
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => ? = 1 - 1
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => ? = 1 - 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => ? = 1 - 1
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => ? = 2 - 1
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => ? = 1 - 1
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => ? = 1 - 1
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => ? = 1 - 1
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => ? = 1 - 1
[2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3 - 1
[2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 2 - 1
[2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 3 - 1
[2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 3 - 1
[2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 5 - 1
[3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 3 - 1
[3,4,2,5,1] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[3,5,2,4,1] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[3,5,4,2,1] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 - 1
[4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[4,2,5,3,1] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[4,3,5,2,1] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[4,5,2,3,1] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2 - 1
[5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1 - 1
[5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[5,4,2,3,1] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1 - 1
[5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001722
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [1,0]
=> 10 => 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => ? = 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 5
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 3
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? = 3
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 3
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 5
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 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: St000091
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [[1],[2]]
=> [2] => 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,2] => 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,1] => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,3] => 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,2,1] => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,2] => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,2] => 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,2,2,2] => ? = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,2,3,1] => ? = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,3,3] => ? = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,3,2,1] => ? = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,4,2] => ? = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,4,2] => ? = 5 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,3,2] => ? = 1 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,1] => ? = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,2,3] => ? = 1 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,2,2,1] => ? = 1 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,3,2] => ? = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,3,2] => ? = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,4] => ? = 1 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,3,1] => ? = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,4] => ? = 1 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,3,1] => ? = 1 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,2,2] => ? = 1 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,2,2] => ? = 1 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 1 - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [3,2,3,2] => ? = 3 - 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [3,3,3,1] => ? = 2 - 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [3,3,2,2] => ? = 3 - 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [3,4,3] => ? = 3 - 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [3,4,3] => ? = 5 - 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,3,2,1] => ? = 1 - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> [4,4,2] => ? = 3 - 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [4,2,3,1] => ? = 1 - 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => ? = 1 - 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,3,3] => ? = 2 - 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,3,3] => ? = 2 - 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [5,4,1] => ? = 1 - 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> [5,3,2] => ? = 2 - 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [5,4,1] => ? = 1 - 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> [5,3,2] => ? = 1 - 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [5,2,3] => ? = 1 - 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [5,2,3] => ? = 1 - 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 1 - 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 2 - 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 1 - 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 1 - 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 1 - 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 1 - 1
Description
The descent variation of a composition.
Defined in [1].
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000709The number of occurrences of 14-2-3 or 14-3-2. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001868The number of alignments of type NE of a signed permutation. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.
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