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Your data matches 209 different statistics following compositions of up to 3 maps.
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Matching statistic: St001344
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001344: 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] => 1
{{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 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: St001847
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001847: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001847: Permutations ⟶ 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] => [3,1,2] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => 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] => [4,2,1,3] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [2,1,3,4] => 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] => [4,3,1,2] => 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] => [3,1,4,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => 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] => [3,2,4,1] => 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] => [2,4,1,3] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => 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] => [5,3,2,1,4] => 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [3,2,1,4,5] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,2,1,3] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,2,1,3] => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,4,2,1,5] => 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] => [5,2,1,3,4] => 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,1,3,4,5] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [4,3,5,1,2] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,5,2,1,4] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => 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] => [5,4,3,1,2] => 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [4,2,3,1,5] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,1,4,5,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,2,3,1,4] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,3,1,2,5] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,2,5] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,3,1,4,2] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,3,1,4,5] => 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] => [4,5,1,2,3] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => 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] => [3,4,1,5,2] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => 0 = 1 - 1
Description
The number of occurrences of the pattern 1432 in a permutation.
Matching statistic: St000036
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000036: 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]]
=> [2,1] => [2,1] => 1
{{1,2,3}}
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St000570
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,2,3] => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,3,2] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,2,3] => 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => [1,2,3] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [1,2,3,4] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => [1,3,2,4] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => [1,2,3,4] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => [1,2,4,3] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => [1,2,3,4] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,5,2,3,4] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,2,3,4,5] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => [1,4,2,3,5] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,4,5,2,3] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,2,3,5,4] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => [1,3,2,4,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,2,3,4,5] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => [1,3,2,4,5] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => [1,3,5,2,4] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,3,2,4,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => [1,3,4,2,5] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,2,4,5,3] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => [1,4,2,3,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [1,5,2,4,3] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [1,2,3,4,5] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => [1,4,5,2,3] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [1,4,2,5,3] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => [1,2,4,3,5] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [1,4,2,3,5] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => [1,4,2,5,3] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,4,5,2,3] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,2,5,3,4] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => [1,5,2,3,4] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,5,2,3,4] => 1
Description
The Edelman-Greene number of a permutation.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
Matching statistic: St000805
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number 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%
Mp00073: Permutations —major-index to inversion-number 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] => [3,1,2] => [3] => 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] => [2,3,1] => [3] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [3,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,4,1] => [2,2] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,4,2,3] => [1,3] => 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] => [2,3,1,4] => [4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,1,3,2] => [3,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [3,4,2,1] => [3,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,4,1] => [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] => [5,1,2,3,4] => [5] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => [4,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,2,3,5,1] => [3,2] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,2,3,1,4] => [3,2] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,4] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,2] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,3,4,2,1] => [1,3,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,5,1,4] => [2,3] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,4,1,5] => [2,3] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,2,1,4,3] => [2,2,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [4,3,5,2,1] => [1,3,1] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,5,1] => [2,3] => 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] => [5,2,1,3,4] => [2,3] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,5,3,4,2] => [1,3,1] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,3] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,5,1,2,3] => [5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,4,2,3,5] => [1,4] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,4,1] => [1,3,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,5,3,4,1] => [4,1] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,3,2,5,1] => [1,2,2] => 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] => [5,3,2,1,4] => [1,2,2] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,5,3,2,4] => [1,2,2] => 1
{{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: St001162
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 1
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [3,2,1] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,2,3] => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [2,1,3] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,1] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,2,4,1] => [1,3,2,4] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [1,4,3,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [4,2,3,1] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [2,1,4,3] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [2,1,3,4] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,2,3,5,1] => [1,3,4,2,5] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,2,5,4,1] => [1,3,2,5,4] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,3,5,4,2] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,2,4,1,5] => [5,3,2,4,1] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,2,5,1,3] => [5,3,1,2,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,2,4,5,1] => [1,3,2,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [3,2,1,4,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,3,4,1] => [1,2,4,5,3] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,4,2,1] => [1,5,3,4,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => [5,2,4,3,1] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,1,2] => [5,1,4,3,2] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,4,3,5,2] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,3,5,1] => [1,2,4,3,5] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,3,2,5,1] => [1,4,3,2,5] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [4,2,3,1,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [4,2,3,5,1] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,5,1,4,2] => [4,1,2,5,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,3,2,1] => [1,5,4,2,3] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,4,1,2,5] => [4,5,2,3,1] => 1
Description
The minimum jump of a permutation.
This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].
Matching statistic: St001496
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001496: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001496: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => ([],2)
=> ([],2)
=> 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
Description
The number of graphs with the same Laplacian spectrum as the given graph.
Matching statistic: St000126
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000126: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000126: 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] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 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] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,[.,.]]]]]}}} in a binary tree.
[[oeis:A036766]] counts binary trees avoiding this pattern.
Matching statistic: St000130
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000130: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000130: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1] => [.,.]
=> 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1] => [.,.]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,1] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [2,3,1] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,2] => [[.,[.,.]],.]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[[.,.],[[.,.],.]]]}}} in a binary tree.
[[oeis:A159771]] counts binary trees avoiding this pattern.
Matching statistic: St000132
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ 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] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 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] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A159773]] counts binary trees avoiding this pattern.
The following 199 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000731The number of double exceedences of a permutation. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000842The breadth of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000516The number of stretching pairs of a permutation. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. 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 $eA$. St000069The number of maximal elements of a poset. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000889The number of alternating sign matrices with the same antidiagonal sums. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St000788The number of nesting-similar perfect matchings of a perfect matching. St001272The number of graphs with the same degree sequence. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001261The Castelnuovo-Mumford regularity of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in 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. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. 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. 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 $K[x]/(x^n)$. 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. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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. 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. St001260The permanent of an alternating sign matrix. St001975The corank of the alternating sign matrix. St000405The number of occurrences of the pattern 1324 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. 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. St000260The radius of a connected graph. St001964The interval resolution global dimension of a poset. 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. 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. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. 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. St001487The number of inner corners of a skew partition. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000315The number of isolated vertices of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000542The number of left-to-right-minima of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001948The number of augmented double ascents of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. 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.
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