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Your data matches 115 different statistics following compositions of up to 3 maps.
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Matching statistic: St000743
St000743: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1 = 0 + 1
[[1],[2]]
=> 1 = 0 + 1
[[1,2,3]]
=> 2 = 1 + 1
[[1,3],[2]]
=> 1 = 0 + 1
[[1,2],[3]]
=> 1 = 0 + 1
[[1],[2],[3]]
=> 2 = 1 + 1
[[1,2,3,4]]
=> 3 = 2 + 1
[[1,3,4],[2]]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 1 = 0 + 1
[[1,2,3],[4]]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> 1 = 0 + 1
[[1,2],[3],[4]]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> 3 = 2 + 1
[[1,2,3,4,5]]
=> 4 = 3 + 1
[[1,3,4,5],[2]]
=> 3 = 2 + 1
[[1,2,4,5],[3]]
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 3 = 2 + 1
[[1,3,5],[2,4]]
=> 2 = 1 + 1
[[1,2,5],[3,4]]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> 1 = 0 + 1
[[1,2,3],[4,5]]
=> 3 = 2 + 1
[[1,4,5],[2],[3]]
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1 = 0 + 1
[[1,2,5],[3],[4]]
=> 2 = 1 + 1
[[1,3,4],[2],[5]]
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> 1 = 0 + 1
[[1,2,3],[4],[5]]
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 3 = 2 + 1
[[1,3],[2,5],[4]]
=> 1 = 0 + 1
[[1,2],[3,5],[4]]
=> 2 = 1 + 1
[[1,3],[2,4],[5]]
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> 4 = 3 + 1
Description
The number of entries in a standard Young tableau such that the next integer is a neighbour.
Matching statistic: St000031
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(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => 3 = 1 + 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St001461
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => 3 = 1 + 2
Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
Matching statistic: St001960
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,1,2] => [2,1,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [4,2,3,1] => 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [4,1,2,3] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [1,4,2,3] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [4,2,1,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => [2,1,4,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => [1,4,3,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => [2,4,3,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => [1,2,3,4] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => [4,3,2,1] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,1,2,3] => [3,4,2,1] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [5,1,2,3,4] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,5,2,3,4] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [5,2,1,3,4] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [5,2,3,1,4] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [2,1,5,3,4] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [1,5,3,2,4] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [2,5,3,1,4] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [5,3,2,1,4] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [5,2,1,4,3] => 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => [2,5,3,4,1] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => [1,2,3,5,4] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,4,1,3,5] => [5,3,2,4,1] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,1,4,5,2] => [1,2,3,4,5] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [2,4,1,5,3] => [1,5,3,4,2] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [2,3,5,1,4] => [5,2,4,3,1] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,3,2,5,1] => [3,2,1,5,4] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,4,5,2,1] => [2,3,1,4,5] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,5,4,3,1] => [1,5,4,3,2] => 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,4,5,1,2] => [2,3,4,1,5] => 0
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5,4,3,2,1] => 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,1,2,3,5] => [3,5,2,4,1] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,5,3] => [3,5,4,1,2] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,5,2,4] => [1,2,4,5,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,5,1,3,4] => [5,4,2,3,1] => 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [4,5,2,3,1] => 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000774
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[[1,2,3]]
=> 00 => [2] => ([],2)
=> 2 = 1 + 1
[[1,3],[2]]
=> 10 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[[1,2],[3]]
=> 01 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[[1],[2],[3]]
=> 11 => [2] => ([],2)
=> 2 = 1 + 1
[[1,2,3,4]]
=> 000 => [3] => ([],3)
=> 3 = 2 + 1
[[1,3,4],[2]]
=> 100 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[1,3],[2,4]]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3,4]]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[1,3],[2],[4]]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> 011 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> 111 => [3] => ([],3)
=> 3 = 2 + 1
[[1,2,3,4,5]]
=> 0000 => [4] => ([],4)
=> 4 = 3 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[[1,2,4,5],[3]]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,2,4],[3,5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,2,4],[3],[5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2,5],[3]]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [4] => ([],4)
=> 4 = 3 + 1
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St001405
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [2,3,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [2,3,4,1] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 3 = 2 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 4 = 3 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 3 = 2 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 3 = 2 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [3,4,5,2,1] => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [3,4,2,5,1] => 1 = 0 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [3,2,4,5,1] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 2 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 3 = 2 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [3,4,2,1,5] => 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => [2,5,4,3,1] => 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [2,5,4,1,3] => 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => [2,3,4,1,5] => 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 3 + 1
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St000010
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,1]
=> 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => [1,1]
=> 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [2,1]
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => [2,1]
=> 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => [2,1,1]
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => [2,1,1]
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => [2,1,1]
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => [3,1]
=> 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => [3,1]
=> 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [2,2,1]
=> 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [4,1]
=> 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => [3,1,1]
=> 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => [2,2,1]
=> 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => [4,1]
=> 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => [3,1,1]
=> 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => [3,1,1]
=> 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => [2,1,1,1]
=> 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => [4,1]
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => [3,1,1]
=> 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => [4,1]
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> 3 = 1 + 2
Description
The length of the partition.
Matching statistic: St000105
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => {{1},{2}}
=> 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000314
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,5,3,2,4] => 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,5,3,4,2] => 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,3,5,4,2] => 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => [1,4,5,2,3] => 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => [1,4,3,2,5] => 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,3,2,5] => 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => [1,3,4,5,2] => 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,5,4,3,2] => 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,4,3,2,5] => 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => [1,5,3,4,2] => 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => [1,3,5,4,2] => 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => [1,4,5,3,2] => 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 3 = 1 + 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000925
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 0 + 2
[[1],[2]]
=> [2,1] => [1,2] => {{1},{2}}
=> 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[[1,3],[2]]
=> [2,1,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 0 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 0 + 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3 = 1 + 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4 = 2 + 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4 = 2 + 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3 = 1 + 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 2 + 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001566The length of the longest arithmetic progression in a permutation. St001330The hat guessing number of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000259The diameter of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001280The number of parts of an integer partition that are at least two. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000260The radius of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001568The smallest positive integer that does not appear twice in the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000478Another weight of a partition according to Alladi. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001867The number of alignments of type EN of a signed permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001820The size of the image of the pop stack sorting operator. St000077The number of boxed and circled entries. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001875The number of simple modules with projective dimension at most 1. St001623The number of doubly irreducible elements of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000906The length of the shortest maximal chain in a poset. St000550The number of modular elements of a lattice. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001857The number of edges in the reduced word graph of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000080The rank of the poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
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