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Matching statistic: St001517
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001517: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001517: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
Description
The length of a longest pair of twins in a permutation.
A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.
Matching statistic: St001569
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001569: 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
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001667
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,4,2,3] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [1,5,2,4,3] => 2
Description
The maximal size of a pair of weak twins for a permutation.
A pair of weak twins in a permutation is a pair of two disjoint subsequences of the same length with the same descent pattern. More formally, a pair of weak twins of size $k$ for a permutation $\pi$ of length $n$ are two disjoint lists $1 \leq i_1 < \dots < i_k \leq n$ and $1 \leq j_1 < \dots < j_k \leq n$ such that $\pi(i_a) < \pi(i_{a+1})$ if and only if $\pi(j_a) < \pi(j_{a+1})$ for all $1 \leq a < k$.
Matching statistic: St000711
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000711: 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] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 1 = 2 - 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001604
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 1 - 2
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 2
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? = 1 - 2
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 2
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 1 - 2
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 2
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 2
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 2 - 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[2,3,4,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[2,4,1,3] => [3,4,1,2] => [2,2]
=> [2]
=> ? = 2 - 2
[2,4,3,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 1 - 2
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[3,2,4,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 1 - 2
[3,4,1,2] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[3,4,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[4,1,2,3] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 1 - 2
[4,1,3,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
[4,2,1,3] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 1 - 2
[4,2,3,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[4,3,1,2] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,2,5,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,3,5,2,4] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,4,2,5,3] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,4,3,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,5,2,3,4] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,5,2,4,3] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
[2,3,5,1,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,1,3,5] => [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,3,5,1] => [5,3,2,4,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,4,5,3,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,1,3,4] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,1,4,3] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,3,1,4] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,3,4,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
[3,1,2,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000264
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00011: Binary trees —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001123
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 1 - 1
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 1
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 1 - 1
[2,3,1] => [2,3,1] => [3]
=> []
=> ? = 1 - 1
[3,1,2] => [3,1,2] => [3]
=> []
=> ? = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 1
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? = 2 - 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 2 - 1
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? = 2 - 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 2 - 1
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? = 2 - 1
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? = 1 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 1 - 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? = 2 - 1
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 1 - 1
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 2 - 1
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 1 - 1
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1 = 2 - 1
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? = 2 - 1
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? = 2 - 1
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? = 2 - 1
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? = 2 - 1
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? = 2 - 1
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? = 2 - 1
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? = 2 - 1
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? = 2 - 1
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? = 2 - 1
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? = 2 - 1
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? = 2 - 1
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? = 2 - 1
[4,1,3,5,2] => [4,1,5,3,2] => [5]
=> []
=> ? = 2 - 1
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St001199
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[2,1] => [2]
=> []
=> []
=> ? = 1 - 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 1 - 1
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[2,3,1] => [3]
=> []
=> []
=> ? = 1 - 1
[3,1,2] => [3]
=> []
=> []
=> ? = 1 - 1
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[2,3,4,1] => [4]
=> []
=> []
=> ? = 2 - 1
[2,4,1,3] => [4]
=> []
=> []
=> ? = 2 - 1
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[3,1,4,2] => [4]
=> []
=> []
=> ? = 1 - 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> []
=> []
=> ? = 2 - 1
[4,1,2,3] => [4]
=> []
=> []
=> ? = 1 - 1
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[4,3,1,2] => [4]
=> []
=> []
=> ? = 2 - 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[2,3,4,5,1] => [5]
=> []
=> []
=> ? = 2 - 1
[2,3,5,1,4] => [5]
=> []
=> []
=> ? = 2 - 1
[2,3,5,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[2,4,1,3,5] => [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[2,4,1,5,3] => [5]
=> []
=> []
=> ? = 2 - 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 2 - 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,5,1,2,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,5,4,1,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,1,5,2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,5,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,3,5,1,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,5,1,3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,5,2,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[5,1,4,3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[5,2,4,3,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[5,3,2,1,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,3,2,4,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[5,3,4,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,4,1,2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,4,2,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,4,3,2,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001060
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[1,2] => [2] => ([],2)
=> ([],2)
=> ? = 1 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([],1)
=> ? = 1 + 1
[1,2,3] => [3] => ([],3)
=> ([],3)
=> ? = 1 + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([],2)
=> ? = 1 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([],2)
=> ? = 1 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> ([],4)
=> ? = 2 + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? = 2 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? = 2 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 2 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? = 1 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 2 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],5)
=> ? = 2 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([],4)
=> ? = 2 + 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? = 2 + 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> ([],4)
=> ? = 2 + 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001603
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 1 - 1
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 1
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 1 - 1
[2,3,1] => [2,3,1] => [3]
=> []
=> ? = 1 - 1
[3,1,2] => [3,1,2] => [3]
=> []
=> ? = 1 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 1
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 2 - 1
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? = 2 - 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 2 - 1
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? = 2 - 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 2 - 1
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? = 2 - 1
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? = 1 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 1 - 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> ? = 2 - 1
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? = 2 - 1
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 1 - 1
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 2 - 1
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 1 - 1
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 2 - 1
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> ? = 2 - 1
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,4,1,2,5] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,4,1,5,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,3,1,5] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,5,3,1,2] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,2,3,4,1] => [5,2,4,3,1] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001605The number of colourings of a cycle such that the multiplicities of colours 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 $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
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