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Your data matches 33 different statistics following compositions of up to 3 maps.
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Matching statistic: St000355
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 1
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000359
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,3,1] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [2,5,4,3,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [5,2,4,3,1] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [5,2,4,3,1] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [5,4,2,3,1] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [4,2,5,3,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [4,5,2,3,1] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [3,2,5,4,1] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [5,3,2,4,1] => 1
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St001745
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1
Description
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000597
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000597: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000472
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => ? = 0
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,1,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [5,4,2,3,1] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,4,2,3,1] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,3,2] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [5,4,1,3,2] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [5,3,2,1,4] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,4,2] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [5,3,4,2,1] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,4,2] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [5,3,4,1,2] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [5,3,4,2,1] => 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [5,2,1,4,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [5,2,4,3,1] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000554
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000554: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000554: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
Description
The number of occurrences of the pattern {{1,2},{3}} in a set partition.
Matching statistic: St000556
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000556: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000556: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} in a set partition.
Matching statistic: St000586
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000586: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Matching statistic: St000599
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block.
Matching statistic: St000605
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001964The interval resolution global dimension of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001811The Castelnuovo-Mumford regularity of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001867The number of alignments of type EN of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car.
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