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Matching statistic: St000842
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Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000842: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 2
[2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,2,3] => 2
[2,1,3] => [1,2,3] => 2
[2,3,1] => [1,2,3] => 2
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 2
[1,2,4,3] => [1,2,3,4] => 2
[1,3,2,4] => [1,2,3,4] => 2
[1,3,4,2] => [1,2,3,4] => 2
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 2
[2,1,4,3] => [1,2,3,4] => 2
[2,3,1,4] => [1,2,3,4] => 2
[2,3,4,1] => [1,2,3,4] => 2
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 2
[1,2,3,5,4] => [1,2,3,4,5] => 2
[1,2,4,3,5] => [1,2,3,4,5] => 2
[1,2,4,5,3] => [1,2,3,4,5] => 2
[1,2,5,3,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => 2
[1,3,2,5,4] => [1,2,3,4,5] => 2
[1,3,4,2,5] => [1,2,3,4,5] => 2
[1,3,4,5,2] => [1,2,3,4,5] => 2
[1,3,5,2,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 2
[1,4,5,3,2] => [1,2,4,3,5] => 2
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Matching statistic: St001162
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1 = 2 - 1
[2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,2,3] => 1 = 2 - 1
[2,1,3] => [1,2,3] => 1 = 2 - 1
[2,3,1] => [1,2,3] => 1 = 2 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => [1,2,3,4] => 1 = 2 - 1
[1,3,2,4] => [1,2,3,4] => 1 = 2 - 1
[1,3,4,2] => [1,2,3,4] => 1 = 2 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 1 = 2 - 1
[2,1,4,3] => [1,2,3,4] => 1 = 2 - 1
[2,3,1,4] => [1,2,3,4] => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 1 = 2 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 1 = 2 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => 1 = 2 - 1
Description
The minimum jump of a permutation.
This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].
Matching statistic: St000767
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1] => 1 = 2 - 1
[2,1] => 0 => [1] => [1] => 1 = 2 - 1
[1,2,3] => 11 => [2] => [1] => 1 = 2 - 1
[1,3,2] => 10 => [1,1] => [2] => 1 = 2 - 1
[2,1,3] => 01 => [1,1] => [2] => 1 = 2 - 1
[2,3,1] => 00 => [2] => [1] => 1 = 2 - 1
[3,1,2] => 00 => [2] => [1] => 1 = 2 - 1
[3,2,1] => 00 => [2] => [1] => 1 = 2 - 1
[1,2,3,4] => 111 => [3] => [1] => 1 = 2 - 1
[1,2,4,3] => 110 => [2,1] => [1,1] => 1 = 2 - 1
[1,3,2,4] => 101 => [1,1,1] => [3] => 1 = 2 - 1
[1,3,4,2] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[1,4,2,3] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[1,4,3,2] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[2,1,3,4] => 011 => [1,2] => [1,1] => 1 = 2 - 1
[2,1,4,3] => 010 => [1,1,1] => [3] => 1 = 2 - 1
[2,3,1,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[2,3,4,1] => 000 => [3] => [1] => 1 = 2 - 1
[2,4,1,3] => 000 => [3] => [1] => 1 = 2 - 1
[2,4,3,1] => 000 => [3] => [1] => 1 = 2 - 1
[3,1,2,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[3,1,4,2] => 000 => [3] => [1] => 1 = 2 - 1
[3,2,1,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[3,2,4,1] => 000 => [3] => [1] => 1 = 2 - 1
[3,4,1,2] => 000 => [3] => [1] => 1 = 2 - 1
[3,4,2,1] => 000 => [3] => [1] => 1 = 2 - 1
[4,1,2,3] => 000 => [3] => [1] => 1 = 2 - 1
[4,1,3,2] => 000 => [3] => [1] => 1 = 2 - 1
[4,2,1,3] => 000 => [3] => [1] => 1 = 2 - 1
[4,2,3,1] => 000 => [3] => [1] => 1 = 2 - 1
[4,3,1,2] => 000 => [3] => [1] => 1 = 2 - 1
[4,3,2,1] => 000 => [3] => [1] => 1 = 2 - 1
[1,2,3,4,5] => 1111 => [4] => [1] => 1 = 2 - 1
[1,2,3,5,4] => 1110 => [3,1] => [1,1] => 1 = 2 - 1
[1,2,4,3,5] => 1101 => [2,1,1] => [1,2] => 2 = 3 - 1
[1,2,4,5,3] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,2,5,3,4] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,2,5,4,3] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,1,2] => [2,1] => 2 = 3 - 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => [4] => 1 = 2 - 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,3,5,2,4] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,3,5,4,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,5,2,3] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,5,3,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
Description
The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Matching statistic: St000903
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1] => 1 = 2 - 1
[2,1] => 0 => [1] => [1] => 1 = 2 - 1
[1,2,3] => 11 => [2] => [1] => 1 = 2 - 1
[1,3,2] => 10 => [1,1] => [2] => 1 = 2 - 1
[2,1,3] => 01 => [1,1] => [2] => 1 = 2 - 1
[2,3,1] => 00 => [2] => [1] => 1 = 2 - 1
[3,1,2] => 00 => [2] => [1] => 1 = 2 - 1
[3,2,1] => 00 => [2] => [1] => 1 = 2 - 1
[1,2,3,4] => 111 => [3] => [1] => 1 = 2 - 1
[1,2,4,3] => 110 => [2,1] => [1,1] => 1 = 2 - 1
[1,3,2,4] => 101 => [1,1,1] => [3] => 1 = 2 - 1
[1,3,4,2] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[1,4,2,3] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[1,4,3,2] => 100 => [1,2] => [1,1] => 1 = 2 - 1
[2,1,3,4] => 011 => [1,2] => [1,1] => 1 = 2 - 1
[2,1,4,3] => 010 => [1,1,1] => [3] => 1 = 2 - 1
[2,3,1,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[2,3,4,1] => 000 => [3] => [1] => 1 = 2 - 1
[2,4,1,3] => 000 => [3] => [1] => 1 = 2 - 1
[2,4,3,1] => 000 => [3] => [1] => 1 = 2 - 1
[3,1,2,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[3,1,4,2] => 000 => [3] => [1] => 1 = 2 - 1
[3,2,1,4] => 001 => [2,1] => [1,1] => 1 = 2 - 1
[3,2,4,1] => 000 => [3] => [1] => 1 = 2 - 1
[3,4,1,2] => 000 => [3] => [1] => 1 = 2 - 1
[3,4,2,1] => 000 => [3] => [1] => 1 = 2 - 1
[4,1,2,3] => 000 => [3] => [1] => 1 = 2 - 1
[4,1,3,2] => 000 => [3] => [1] => 1 = 2 - 1
[4,2,1,3] => 000 => [3] => [1] => 1 = 2 - 1
[4,2,3,1] => 000 => [3] => [1] => 1 = 2 - 1
[4,3,1,2] => 000 => [3] => [1] => 1 = 2 - 1
[4,3,2,1] => 000 => [3] => [1] => 1 = 2 - 1
[1,2,3,4,5] => 1111 => [4] => [1] => 1 = 2 - 1
[1,2,3,5,4] => 1110 => [3,1] => [1,1] => 1 = 2 - 1
[1,2,4,3,5] => 1101 => [2,1,1] => [1,2] => 2 = 3 - 1
[1,2,4,5,3] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,2,5,3,4] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,2,5,4,3] => 1100 => [2,2] => [2] => 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,1,2] => [2,1] => 2 = 3 - 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => [4] => 1 = 2 - 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,3,5,2,4] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,3,5,4,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,1,1] => 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,5,2,3] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
[1,4,5,3,2] => 1000 => [1,3] => [1,1] => 1 = 2 - 1
Description
The number of different parts of an integer composition.
Matching statistic: St001220
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [3,2,1] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[3,1,2] => [3,2,1] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1 = 2 - 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1 = 2 - 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1 = 2 - 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,5,3] => 2 = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,5,3] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,5,3] => 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
Description
The width of a permutation.
Let $w$ be a permutation. The interval $[e,w]$ in the weak order is ranked, and we define $r_i=r_i(w)$ to be the number of elements at rank $i$ in $[e,w]$, where $i \in \{0, \dots, \ell(w)\}$. The ''width'' of
$w$ is the maximum of $\{r_0,r_1,\ldots,r_{\ell(w)}\}$. See [1].
Matching statistic: St001344
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => [1,2] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,3,2] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[2,3,1,4] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St001518
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> 1 = 2 - 1
[2,1] => [1,2] => [1,2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 2 - 1
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 2 - 1
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The number of graphs with the same ordinary spectrum as the given graph.
Matching statistic: St000516
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 2 - 2
[2,1] => [2,1] => [1,2] => [1,2] => 0 = 2 - 2
[1,2,3] => [1,3,2] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[3,2,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0 = 2 - 2
[1,2,3,4] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,2,4,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,3,2,4] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[2,1,3,4] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,3,1,4] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[2,4,1,3] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0 = 2 - 2
[3,1,2,4] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 0 = 2 - 2
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 0 = 2 - 2
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[3,2,4,1] => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 0 = 2 - 2
[3,4,1,2] => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 0 = 2 - 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,3,1,2] => [4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 0 = 2 - 2
[1,2,3,4,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 2 - 2
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000666
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 2 - 2
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 2 - 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 0 = 2 - 2
[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 0 = 2 - 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 0 = 2 - 2
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [2,1,4,3] => 0 = 2 - 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 0 = 2 - 2
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [2,1,4,3] => 0 = 2 - 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 0 = 2 - 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 0 = 2 - 2
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 0 = 2 - 2
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 0 = 2 - 2
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 0 = 2 - 2
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 0 = 2 - 2
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 0 = 2 - 2
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 0 = 2 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 2 - 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 2 - 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => [1,3,2,5,4] => 0 = 2 - 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 2 - 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,1,2,4] => [1,3,2,5,4] => 0 = 2 - 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 2 - 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 2 - 2
Description
The number of right tethers of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right tether is a large ascent between two consecutive rafts of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001673
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1] => 0 = 2 - 2
[2,1] => 0 => [1] => [1] => 0 = 2 - 2
[1,2,3] => 11 => [2] => [1] => 0 = 2 - 2
[1,3,2] => 10 => [1,1] => [2] => 0 = 2 - 2
[2,1,3] => 01 => [1,1] => [2] => 0 = 2 - 2
[2,3,1] => 00 => [2] => [1] => 0 = 2 - 2
[3,1,2] => 00 => [2] => [1] => 0 = 2 - 2
[3,2,1] => 00 => [2] => [1] => 0 = 2 - 2
[1,2,3,4] => 111 => [3] => [1] => 0 = 2 - 2
[1,2,4,3] => 110 => [2,1] => [1,1] => 0 = 2 - 2
[1,3,2,4] => 101 => [1,1,1] => [3] => 0 = 2 - 2
[1,3,4,2] => 100 => [1,2] => [1,1] => 0 = 2 - 2
[1,4,2,3] => 100 => [1,2] => [1,1] => 0 = 2 - 2
[1,4,3,2] => 100 => [1,2] => [1,1] => 0 = 2 - 2
[2,1,3,4] => 011 => [1,2] => [1,1] => 0 = 2 - 2
[2,1,4,3] => 010 => [1,1,1] => [3] => 0 = 2 - 2
[2,3,1,4] => 001 => [2,1] => [1,1] => 0 = 2 - 2
[2,3,4,1] => 000 => [3] => [1] => 0 = 2 - 2
[2,4,1,3] => 000 => [3] => [1] => 0 = 2 - 2
[2,4,3,1] => 000 => [3] => [1] => 0 = 2 - 2
[3,1,2,4] => 001 => [2,1] => [1,1] => 0 = 2 - 2
[3,1,4,2] => 000 => [3] => [1] => 0 = 2 - 2
[3,2,1,4] => 001 => [2,1] => [1,1] => 0 = 2 - 2
[3,2,4,1] => 000 => [3] => [1] => 0 = 2 - 2
[3,4,1,2] => 000 => [3] => [1] => 0 = 2 - 2
[3,4,2,1] => 000 => [3] => [1] => 0 = 2 - 2
[4,1,2,3] => 000 => [3] => [1] => 0 = 2 - 2
[4,1,3,2] => 000 => [3] => [1] => 0 = 2 - 2
[4,2,1,3] => 000 => [3] => [1] => 0 = 2 - 2
[4,2,3,1] => 000 => [3] => [1] => 0 = 2 - 2
[4,3,1,2] => 000 => [3] => [1] => 0 = 2 - 2
[4,3,2,1] => 000 => [3] => [1] => 0 = 2 - 2
[1,2,3,4,5] => 1111 => [4] => [1] => 0 = 2 - 2
[1,2,3,5,4] => 1110 => [3,1] => [1,1] => 0 = 2 - 2
[1,2,4,3,5] => 1101 => [2,1,1] => [1,2] => 1 = 3 - 2
[1,2,4,5,3] => 1100 => [2,2] => [2] => 0 = 2 - 2
[1,2,5,3,4] => 1100 => [2,2] => [2] => 0 = 2 - 2
[1,2,5,4,3] => 1100 => [2,2] => [2] => 0 = 2 - 2
[1,3,2,4,5] => 1011 => [1,1,2] => [2,1] => 1 = 3 - 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [4] => 0 = 2 - 2
[1,3,4,2,5] => 1001 => [1,2,1] => [1,1,1] => 0 = 2 - 2
[1,3,4,5,2] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,3,5,2,4] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,3,5,4,2] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,4,2,3,5] => 1001 => [1,2,1] => [1,1,1] => 0 = 2 - 2
[1,4,2,5,3] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,4,3,2,5] => 1001 => [1,2,1] => [1,1,1] => 0 = 2 - 2
[1,4,3,5,2] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,4,5,2,3] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
[1,4,5,3,2] => 1000 => [1,3] => [1,1] => 0 = 2 - 2
Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
The following 309 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001282The number of graphs with the same chromatic polynomial. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000068The number of minimal elements in a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000878The number of ones minus the number of zeros of a binary word. St000259The diameter of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001060The distinguishing index of a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001651The Frankl number of a lattice. St000260The radius of a connected graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000741The Colin de Verdière graph invariant. St001890The maximum magnitude of the Möbius function of a poset. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000408The number of occurrences of the pattern 4231 in a permutation. St000322The skewness of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001490The number of connected components of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001330The hat guessing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000326The position of the first one in a binary word after appending a 1 at the end. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St000297The number of leading ones in a binary word. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001624The breadth of a lattice. St000667The greatest common divisor of the parts of the partition. St000570The Edelman-Greene number of a permutation. St000958The number of Bruhat factorizations of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001555The order of a signed permutation. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St000478Another weight of a partition according to Alladi. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001568The smallest positive integer that does not appear twice in the partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St000069The number of maximal elements of a poset. St000629The defect of a binary word. St001429The number of negative entries in a signed permutation. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001889The size of the connectivity set of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001616The number of neutral elements in a lattice. St000627The exponent of a binary word. St000731The number of double exceedences of a permutation. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001863The number of weak excedances of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000098The chromatic number of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001395The number of strictly unfriendly partitions of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000258The burning number of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001430The number of positive entries in a signed permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001691The number of kings in a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000328The maximum number of child nodes in a tree. St000876The number of factors in the Catalan decomposition of a binary word. St000905The number of different multiplicities of parts of an integer composition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001884The number of borders of a binary word. St000264The girth of a graph, which is not a tree. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001410The minimal entry of a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001979The size of the permutation set corresponding to the alternating sign matrix variety. St000074The number of special entries. St000366The number of double descents of a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000894The trace of an alternating sign matrix. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001335The cardinality of a minimal cycle-isolating set of a graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001975The corank of the alternating sign matrix. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000907The number of maximal antichains of minimal length in a poset. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001765The number of connected components of the friends and strangers graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
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