- FindStat

Your data matches 115 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001162
(load all 14 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => 1
{{1},{2}}
 => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => 1
{{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => 1
{{1,2,3,4}}
 => [2,3,4,1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 1

Description

The minimum jump of a permutation. This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].

Matching statistic: St000842
(load all 14 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => 2 = 1 + 1
{{1},{2}}
 => [1,2] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => 2 = 1 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 2 = 1 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 2 = 1 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2 = 1 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 2 = 1 + 1

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: St000805

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => [2] => 1
{{1},{2}}
 => [1,2] => [1,2] => [2] => 1
{{1,2,3}}
 => [2,3,1] => [1,3,2] => [1,2] => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [3] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,4,3] => [2,2] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,3,2,4] => [1,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => [3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,2,3] => [1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => [2,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [2,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [1,2,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => [4] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => [4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => [3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => [1,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => [4] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,5,4] => [3,2] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,4,3,5] => [2,3] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => [4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,3,4] => [2,3] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,3,2,4,5] => [1,4] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,1,2,5,4] => [3,2] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [3,2] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => [3,2] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,1,3,2] => [1,3,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,3,5,2,4] => [1,4] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,2,3,5] => [1,4] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,5,1,4,2] => [4,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,1,4,2,3] => [2,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,5,2,3,4] => [1,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,5,4] => [3,2] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,4,5,3,2] => [1,3,1] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,4,3,5] => [2,3] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,5,4,3] => [2,2,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,5,3,2,4] => [1,2,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [2,3] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,5,2,4,3] => [1,3,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,5,3,4,2] => [1,3,1] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,3,4] => [2,3] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [2,3] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,5,4] => [2,1,2] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,2,1,5,3] => [2,1,2] => 2
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,2,2] => 1

Description

The number of peaks of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

Matching statistic: St000905

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000905: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2] => [1] => 1
{{1},{2}}
 => [1,2] => [2] => [1] => 1
{{1,2,3}}
 => [2,3,1] => [3] => [1] => 1
{{1,2},{3}}
 => [2,1,3] => [3] => [1] => 1
{{1,3},{2}}
 => [3,2,1] => [2,1] => [1,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2] => [1,1] => 1
{{1},{2},{3}}
 => [1,2,3] => [3] => [1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4] => [1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [4] => [1] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => [1,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,2] => [2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4] => [1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,2] => [2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4] => [1] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,2] => [2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3] => [1,1] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3] => [1,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => [1,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,2] => [2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4] => [1] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5] => [1] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5] => [1] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => [1,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2] => [1,1] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [5] => [1] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,2] => [1,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5] => [1] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,2] => [1,1] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => [2,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,3] => [1,1] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,3] => [1,1] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => [1,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => [2,1] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2] => [1,1] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [5] => [1] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,3] => [1,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [2,3] => [1,1] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,3] => [1,1] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => [1,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2] => [1,1] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [5] => [1] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => [2,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => [1,1] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,2] => [1,1,1] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3] => [1,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,2,2] => [1,2] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4] => [1,1] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,2,2] => [1,2] => 1

Description

The number of different multiplicities of parts of an integer composition.

Matching statistic: St001208
(load all 55 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => [1] => 1
{{1},{2}}
 => [1,2] => [1,2] => [1] => 1
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1] => 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => [2,1] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [1,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => [2,1,3] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [2,3,1] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => [2,3,1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [3,2,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [3,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => [3,2,1] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => [3,1,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => [3,1,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => [1,2,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,1,3,4] => [2,1,3,4] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,3,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => [4,2,3,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,1,3] => [4,2,1,3] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,3] => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,5,3] => [2,4,1,3] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,2,3,1,4] => [2,3,1,4] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,1] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [3,2,4,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => [3,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => [3,4,2,1] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,4] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,4,2,5] => [3,1,4,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,1,2] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,1,5,4] => [3,2,1,4] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,3,4,1,2] => [3,4,1,2] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,5,4,1] => [3,2,4,1] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => [3,4,2,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,5,2,1] => [4,3,2,1] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,1,3,2] => [4,1,3,2] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,5,3,2,1] => [4,3,2,1] => 1

Description

The 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 $K[x]/(x^n)$.

Matching statistic: St001344
(load all 11 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,3,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => [4,2,1,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,1,2,4] => [3,1,4,2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,1,2,3] => [4,1,3,2] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,2,3] => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,4,3,1] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [4,2,5,1,3] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,3,1,4] => [5,2,4,1,3] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,2,5,4,1] => [3,2,5,4,1] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,1,3,5] => [4,2,1,5,3] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,2,1,3,4] => [5,2,1,4,3] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,2,4,1,3] => [5,2,4,1,3] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [2,5,3,1,4] => [2,5,4,1,3] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,1,3,4,2] => [5,1,4,3,2] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,5,3,1,2] => [4,5,3,1,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [2,4,3,1,5] => [2,5,4,1,3] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,3,2,1,4] => [5,3,2,1,4] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,5,3,2,1] => [4,5,3,2,1] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,5,4,3,1] => [2,5,4,3,1] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,5,3,2] => 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: St001493
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => [1,1,0,0]
 => 1
{{1},{2}}
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1

Description

The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.

Matching statistic: St000129

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000129: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => [[.,.],.]
 => 0 = 1 - 1
{{1},{2}}
 => [1,2] => [1,2] => [.,[.,.]]
 => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [[[.,.],.],.]
 => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0 = 1 - 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => 0 = 1 - 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[[.,.],.],.]]]}}} in a binary tree. [[oeis:A159770]] counts binary trees avoiding this pattern.

Matching statistic: St000132

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [1,2] => [.,[.,.]]
 => 0 = 1 - 1
{{1},{2}}
 => [1,2] => [2,1] => [[.,.],.]
 => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [[.,.],[.,.]]
 => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => [[.,.],[.,.]]
 => 0 = 1 - 1
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [[.,[.,.]],.]
 => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [[[.,.],.],.]
 => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0 = 1 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0 = 1 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,2,1,5,3] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1,3,2,5] => [[.,[[.,.],.]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => 0 = 1 - 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => 0 = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => 0 = 1 - 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => 1 = 2 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => 0 = 1 - 1

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A159773]] counts binary trees avoiding this pattern.

Matching statistic: St000449

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000449: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
{{1},{2}}
 => [1,2] => ([],2)
 => ([],2)
 => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => ([],3)
 => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 1 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0 = 1 - 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1

Description

The number of pairs of vertices of a graph with distance 4. This is the coefficient of the quartic term of the Wiener polynomial.

The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001550The number of inversions between exceedances where the greater exceedance is linked. St001271The competition number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St000741The Colin de Verdière graph invariant. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000284The Plancherel distribution on integer partitions. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001890The maximum magnitude of the Möbius function of a poset. St001866The nesting alignments of a signed permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001845The number of join irreducibles minus the rank of a lattice. 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. St001429The number of negative entries in a signed permutation. St000068The number of minimal elements in a poset. St001862The number of crossings of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset.