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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000882
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000882: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 1
[-] => [1] => 1
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [1,2] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,2,3] => 1
[-,3,2] => [2,1,3] => 1
[2,1,+] => [1,3,2] => 1
[2,1,-] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,2,3] => 1
[3,+,1] => [2,1,3] => 1
[3,-,1] => [1,3,2] => 1
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 1
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,3,4] => 1
[-,+,4,3] => [2,3,1,4] => 1
[+,-,4,3] => [1,3,2,4] => 1
[-,-,4,3] => [3,1,2,4] => 1
[+,3,2,+] => [1,2,4,3] => 1
[-,3,2,+] => [2,4,1,3] => 1
[+,3,2,-] => [1,2,3,4] => 1
[-,3,2,-] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => 1
[-,3,4,2] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => 1
Description
The number of connected components of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number connected components of such short braid moves among all reduced words.
Matching statistic: St000119
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0 = 1 - 1
[-] => [1] => 0 = 1 - 1
[+,+] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => 0 = 1 - 1
[+,-,+] => [1,3,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => 0 = 1 - 1
[2,1,+] => [1,3,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => 0 = 1 - 1
[3,-,1] => [1,3,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => 0 = 1 - 1
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000002
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [2,1] => 0 = 1 - 1
[-,+] => [2,1] => [1,2] => 0 = 1 - 1
[+,-] => [1,2] => [2,1] => 0 = 1 - 1
[-,-] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1,3] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000095
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 0 = 1 - 1
[-] => [1] => ([],1)
=> 0 = 1 - 1
[+,+] => [1,2] => ([],2)
=> 0 = 1 - 1
[-,+] => [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
[+,-] => [1,2] => ([],2)
=> 0 = 1 - 1
[-,-] => [1,2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,2] => ([],2)
=> 0 = 1 - 1
[+,+,+] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[+,-,+] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[+,+,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[-,+,-] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[+,-,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,-,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[+,3,2] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,3,2] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1,+] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[3,+,1] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[3,-,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
Description
The number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001328
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 0 = 1 - 1
[-] => [1] => ([],1)
=> 0 = 1 - 1
[+,+] => [1,2] => ([],2)
=> 0 = 1 - 1
[-,+] => [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
[+,-] => [1,2] => ([],2)
=> 0 = 1 - 1
[-,-] => [1,2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,2] => ([],2)
=> 0 = 1 - 1
[+,+,+] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[+,-,+] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[+,+,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[-,+,-] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[+,-,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,-,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[+,3,2] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[-,3,2] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1,+] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,1,-] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[3,+,1] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[3,-,1] => [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001396
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 0 = 1 - 1
[-] => [1] => ([],1)
=> 0 = 1 - 1
[+,+] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[-,+] => [2,1] => ([],2)
=> 0 = 1 - 1
[+,-] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[-,-] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[-,+,+] => [2,3,1] => ([(1,2)],3)
=> 0 = 1 - 1
[+,-,+] => [1,3,2] => ([(0,1),(0,2)],3)
=> 0 = 1 - 1
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[-,-,+] => [3,1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[-,+,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[+,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[-,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,1,+] => [1,3,2] => ([(0,1),(0,2)],3)
=> 0 = 1 - 1
[2,1,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,+,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,-,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
Description
Number of triples of incomparable elements in a finite poset.
For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.
Matching statistic: St000223
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,+] => [2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,1,-] => [2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [2,3,1] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,2,1] => [1,3,2] => 0 = 1 - 1
[3,+,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,-,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[+,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[-,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[+,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
Description
The number of nestings in the permutation.
Matching statistic: St000232
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => {{1}}
=> {{1}}
=> 0 = 1 - 1
[-] => [1] => {{1}}
=> {{1}}
=> 0 = 1 - 1
[+,+] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 0 = 1 - 1
[-,+] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 0 = 1 - 1
[+,-] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 0 = 1 - 1
[-,-] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 0 = 1 - 1
[2,1] => [2,1] => {{1,2}}
=> {{1},{2}}
=> 0 = 1 - 1
[+,+,+] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[-,+,+] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[+,-,+] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[+,+,-] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[-,-,+] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[-,+,-] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[+,-,-] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[-,-,-] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 0 = 1 - 1
[+,3,2] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 0 = 1 - 1
[-,3,2] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 0 = 1 - 1
[2,1,+] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0 = 1 - 1
[2,1,-] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0 = 1 - 1
[2,3,1] => [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0 = 1 - 1
[3,1,2] => [3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> 0 = 1 - 1
[3,+,1] => [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 0 = 1 - 1
[3,-,1] => [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 0 = 1 - 1
[-,+,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 0 = 1 - 1
[+,-,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 0 = 1 - 1
[-,-,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 0 = 1 - 1
[+,3,2,+] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[-,3,2,+] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[+,3,2,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[-,3,2,-] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[+,3,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 0 = 1 - 1
[-,3,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 0 = 1 - 1
[+,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 0 = 1 - 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [3,1,2] => [2,3,1] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000371
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 0 = 1 - 1
[-] => [1] => [1] => [1] => 0 = 1 - 1
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,+] => [2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,1,-] => [2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [2,3,1] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,2,1] => [1,3,2] => 0 = 1 - 1
[3,+,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,-,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[+,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[-,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[+,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001718The number of non-empty open intervals in a poset. St001082The number of boxed occurrences of 123 in a permutation. St000516The number of stretching pairs of a permutation. St000649The number of 3-excedences of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001490The number of connected components of a skew partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000908The length of the shortest maximal antichain in a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000181The number of connected components of the Hasse diagram for the poset. St001964The interval resolution global dimension of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000456The monochromatic index of a connected graph. St000907The number of maximal antichains of minimal length in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001926Sparre Andersen's position of the maximum of a signed permutation.
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