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Your data matches 132 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [2] => 2 = 3 - 1
[2,1] => [1,1] => 1 = 2 - 1
[1,2,3] => [3] => 3 = 4 - 1
[1,3,2] => [2,1] => 2 = 3 - 1
[2,1,3] => [1,2] => 1 = 2 - 1
[2,3,1] => [2,1] => 2 = 3 - 1
[3,1,2] => [1,2] => 1 = 2 - 1
[3,2,1] => [1,1,1] => 1 = 2 - 1
[1,2,3,4] => [4] => 4 = 5 - 1
[1,2,4,3] => [3,1] => 3 = 4 - 1
[1,3,2,4] => [2,2] => 2 = 3 - 1
[1,3,4,2] => [3,1] => 3 = 4 - 1
[1,4,2,3] => [2,2] => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => 2 = 3 - 1
[2,1,3,4] => [1,3] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => 1 = 2 - 1
[2,3,1,4] => [2,2] => 2 = 3 - 1
[2,3,4,1] => [3,1] => 3 = 4 - 1
[2,4,1,3] => [2,2] => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => 2 = 3 - 1
[3,1,2,4] => [1,3] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => 1 = 2 - 1
[3,4,1,2] => [2,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => 2 = 3 - 1
[4,1,2,3] => [1,3] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => 1 = 2 - 1
[1,2,3,4,5] => [5] => 5 = 6 - 1
[1,2,3,5,4] => [4,1] => 4 = 5 - 1
[1,2,4,3,5] => [3,2] => 3 = 4 - 1
[1,2,4,5,3] => [4,1] => 4 = 5 - 1
[1,2,5,3,4] => [3,2] => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => 3 = 4 - 1
[1,3,2,4,5] => [2,3] => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => 3 = 4 - 1
[1,3,4,5,2] => [4,1] => 4 = 5 - 1
[1,3,5,2,4] => [3,2] => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => 3 = 4 - 1
[1,4,2,3,5] => [2,3] => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => 3 = 4 - 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [2] => 2 = 3 - 1
[2,1] => [1,1] => 1 = 2 - 1
[1,2,3] => [3] => 3 = 4 - 1
[1,3,2] => [2,1] => 1 = 2 - 1
[2,1,3] => [1,2] => 2 = 3 - 1
[2,3,1] => [2,1] => 1 = 2 - 1
[3,1,2] => [1,2] => 2 = 3 - 1
[3,2,1] => [1,1,1] => 1 = 2 - 1
[1,2,3,4] => [4] => 4 = 5 - 1
[1,2,4,3] => [3,1] => 1 = 2 - 1
[1,3,2,4] => [2,2] => 2 = 3 - 1
[1,3,4,2] => [3,1] => 1 = 2 - 1
[1,4,2,3] => [2,2] => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => 1 = 2 - 1
[2,1,3,4] => [1,3] => 3 = 4 - 1
[2,1,4,3] => [1,2,1] => 1 = 2 - 1
[2,3,1,4] => [2,2] => 2 = 3 - 1
[2,3,4,1] => [3,1] => 1 = 2 - 1
[2,4,1,3] => [2,2] => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => 1 = 2 - 1
[3,1,2,4] => [1,3] => 3 = 4 - 1
[3,1,4,2] => [1,2,1] => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 2 = 3 - 1
[3,2,4,1] => [1,2,1] => 1 = 2 - 1
[3,4,1,2] => [2,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => 1 = 2 - 1
[4,1,2,3] => [1,3] => 3 = 4 - 1
[4,1,3,2] => [1,2,1] => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 2 = 3 - 1
[4,2,3,1] => [1,2,1] => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => 1 = 2 - 1
[1,2,3,4,5] => [5] => 5 = 6 - 1
[1,2,3,5,4] => [4,1] => 1 = 2 - 1
[1,2,4,3,5] => [3,2] => 2 = 3 - 1
[1,2,4,5,3] => [4,1] => 1 = 2 - 1
[1,2,5,3,4] => [3,2] => 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => 3 = 4 - 1
[1,3,2,5,4] => [2,2,1] => 1 = 2 - 1
[1,3,4,2,5] => [3,2] => 2 = 3 - 1
[1,3,4,5,2] => [4,1] => 1 = 2 - 1
[1,3,5,2,4] => [3,2] => 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 3 = 4 - 1
[1,4,2,5,3] => [2,2,1] => 1 = 2 - 1
[1,4,3,2,5] => [2,1,2] => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => 1 = 2 - 1
[1,4,5,2,3] => [3,2] => 2 = 3 - 1
Description
The last part of an integer composition.
Matching statistic: St000745
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 1 = 2 - 1
[1,2] => [[1,2]]
=> 1 = 2 - 1
[2,1] => [[1],[2]]
=> 2 = 3 - 1
[1,2,3] => [[1,2,3]]
=> 1 = 2 - 1
[1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[2,1,3] => [[1,3],[2]]
=> 2 = 3 - 1
[2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[3,1,2] => [[1,3],[2]]
=> 2 = 3 - 1
[3,2,1] => [[1],[2],[3]]
=> 3 = 4 - 1
[1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,3,2] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[2,1,3,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[2,3,4,1] => [[1,2,3],[4]]
=> 1 = 2 - 1
[2,4,1,3] => [[1,2],[3,4]]
=> 1 = 2 - 1
[2,4,3,1] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[3,1,2,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[3,1,4,2] => [[1,3],[2,4]]
=> 2 = 3 - 1
[3,2,1,4] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[3,4,2,1] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[4,1,2,3] => [[1,3,4],[2]]
=> 2 = 3 - 1
[4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[4,2,1,3] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[4,3,1,2] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 5 - 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 2 - 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1 = 2 - 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1 = 2 - 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 1 = 2 - 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001050
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 1 = 2 - 1
[1,2] => {{1},{2}}
=> 2 = 3 - 1
[2,1] => {{1,2}}
=> 1 = 2 - 1
[1,2,3] => {{1},{2},{3}}
=> 3 = 4 - 1
[1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[2,1,3] => {{1,2},{3}}
=> 2 = 3 - 1
[2,3,1] => {{1,2,3}}
=> 1 = 2 - 1
[3,1,2] => {{1,2,3}}
=> 1 = 2 - 1
[3,2,1] => {{1,3},{2}}
=> 2 = 3 - 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 5 - 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 2 = 3 - 1
[1,3,4,2] => {{1},{2,3,4}}
=> 1 = 2 - 1
[1,4,2,3] => {{1},{2,3,4}}
=> 1 = 2 - 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 4 - 1
[2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[2,3,1,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[2,3,4,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[2,4,1,3] => {{1,2,3,4}}
=> 1 = 2 - 1
[2,4,3,1] => {{1,2,4},{3}}
=> 2 = 3 - 1
[3,1,2,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[3,1,4,2] => {{1,2,3,4}}
=> 1 = 2 - 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 4 - 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1 = 2 - 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2 = 3 - 1
[3,4,2,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,1,2,3] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,1,3,2] => {{1,2,4},{3}}
=> 2 = 3 - 1
[4,2,1,3] => {{1,3,4},{2}}
=> 1 = 2 - 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 4 - 1
[4,3,1,2] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,3,2,1] => {{1,4},{2,3}}
=> 2 = 3 - 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 6 - 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2 = 3 - 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3 = 4 - 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2 = 3 - 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3 = 4 - 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2 = 3 - 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St001051
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001051: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 1 = 2 - 1
[1,2] => {{1},{2}}
=> 2 = 3 - 1
[2,1] => {{1,2}}
=> 1 = 2 - 1
[1,2,3] => {{1},{2},{3}}
=> 3 = 4 - 1
[1,3,2] => {{1},{2,3}}
=> 2 = 3 - 1
[2,1,3] => {{1,2},{3}}
=> 2 = 3 - 1
[2,3,1] => {{1,2,3}}
=> 1 = 2 - 1
[3,1,2] => {{1,2,3}}
=> 1 = 2 - 1
[3,2,1] => {{1,3},{2}}
=> 1 = 2 - 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 5 - 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 2 = 3 - 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 4 - 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2 = 3 - 1
[1,4,2,3] => {{1},{2,3,4}}
=> 2 = 3 - 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 4 - 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 4 - 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[2,3,1,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[2,3,4,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[2,4,1,3] => {{1,2,3,4}}
=> 1 = 2 - 1
[2,4,3,1] => {{1,2,4},{3}}
=> 1 = 2 - 1
[3,1,2,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[3,1,4,2] => {{1,2,3,4}}
=> 1 = 2 - 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 3 - 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1 = 2 - 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2 = 3 - 1
[3,4,2,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,1,2,3] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,1,3,2] => {{1,2,4},{3}}
=> 1 = 2 - 1
[4,2,1,3] => {{1,3,4},{2}}
=> 1 = 2 - 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[4,3,1,2] => {{1,2,3,4}}
=> 1 = 2 - 1
[4,3,2,1] => {{1,4},{2,3}}
=> 1 = 2 - 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 6 - 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 3 = 4 - 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 3 = 4 - 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 5 - 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2 = 3 - 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3 = 4 - 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2 = 3 - 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 2 = 3 - 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 4 - 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 4 - 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 2 = 3 - 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 5 - 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 2 = 3 - 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3 = 4 - 1
Description
The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.
The bijection between set partitions of $\{1,\dots,n\}$ into $k$ blocks and trees with $n+1-k$ leaves is described in Theorem 1 of [1].
Matching statistic: St000439
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 2
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 3
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 4
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000025
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2 = 3 - 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => [2] => [1,1,0,0]
=> 2 = 3 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000297
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1 = 2 - 1
[1,2] => [2] => 10 => 1 = 2 - 1
[2,1] => [1,1] => 11 => 2 = 3 - 1
[1,2,3] => [3] => 100 => 1 = 2 - 1
[1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3] => [1,2] => 110 => 2 = 3 - 1
[2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2] => [1,2] => 110 => 2 = 3 - 1
[3,2,1] => [1,1,1] => 111 => 3 = 4 - 1
[1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,3,4] => [1,3] => 1100 => 2 = 3 - 1
[2,1,4,3] => [1,2,1] => 1101 => 2 = 3 - 1
[2,3,1,4] => [2,2] => 1010 => 1 = 2 - 1
[2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[2,4,1,3] => [2,2] => 1010 => 1 = 2 - 1
[2,4,3,1] => [2,1,1] => 1011 => 1 = 2 - 1
[3,1,2,4] => [1,3] => 1100 => 2 = 3 - 1
[3,1,4,2] => [1,2,1] => 1101 => 2 = 3 - 1
[3,2,1,4] => [1,1,2] => 1110 => 3 = 4 - 1
[3,2,4,1] => [1,2,1] => 1101 => 2 = 3 - 1
[3,4,1,2] => [2,2] => 1010 => 1 = 2 - 1
[3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[4,1,2,3] => [1,3] => 1100 => 2 = 3 - 1
[4,1,3,2] => [1,2,1] => 1101 => 2 = 3 - 1
[4,2,1,3] => [1,1,2] => 1110 => 3 = 4 - 1
[4,2,3,1] => [1,2,1] => 1101 => 2 = 3 - 1
[4,3,1,2] => [1,1,2] => 1110 => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1] => 1111 => 4 = 5 - 1
[1,2,3,4,5] => [5] => 10000 => 1 = 2 - 1
[1,2,3,5,4] => [4,1] => 10001 => 1 = 2 - 1
[1,2,4,3,5] => [3,2] => 10010 => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => 10001 => 1 = 2 - 1
[1,2,5,3,4] => [3,2] => 10010 => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => 10100 => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => 10101 => 1 = 2 - 1
[1,3,4,2,5] => [3,2] => 10010 => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => 10001 => 1 = 2 - 1
[1,3,5,2,4] => [3,2] => 10010 => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => 10011 => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 10100 => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => 10101 => 1 = 2 - 1
[1,4,3,2,5] => [2,1,2] => 10110 => 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => 10101 => 1 = 2 - 1
[1,4,5,2,3] => [3,2] => 10010 => 1 = 2 - 1
Description
The number of leading ones in a binary word.
The following 122 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000010The length of the partition. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000759The smallest missing part in an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000053The number of valleys of the Dyck path. St001176The size of a partition minus its first part. St000678The number of up steps after the last double rise of a Dyck path. St000287The number of connected components of a graph. St000504The cardinality of the first block of a set partition. St000553The number of blocks of a graph. St000675The number of centered multitunnels of a Dyck path. St000823The number of unsplittable factors of the set partition. St001316The domatic number of a graph. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St001828The Euler characteristic of a graph. St000234The number of global ascents of a permutation. St000617The number of global maxima of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000989The number of final rises of a permutation. St000068The number of minimal elements in a poset. St000717The number of ordinal summands of a poset. St000286The number of connected components of the complement of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000993The multiplicity of the largest part of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000054The first entry of the permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001118The acyclic chromatic index of a graph. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000015The number of peaks of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000237The number of small exceedances. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001060The distinguishing index of a graph. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St000181The number of connected components of the Hasse diagram for the poset. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000454The largest eigenvalue of a graph if it is integral. St000806The semiperimeter of the associated bargraph. St000843The decomposition number of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St000264The girth of a graph, which is not a tree. St000740The last entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000031The number of cycles in the cycle decomposition of a permutation. St000738The first entry in the last row of a standard tableau. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000258The burning number of a graph. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000335The difference of lower and upper interactions. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000133The "bounce" of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St001340The cardinality of a minimal non-edge isolating set of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000338The number of pixed points of a permutation. St000732The number of double deficiencies of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001862The number of crossings of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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