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Matching statistic: St000507
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000010
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [4,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [4,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,1]
=> 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,9,2,8,7,6,5,4,1] => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,7,8,9,1] => [5,4,3,2,9,8,7,6,1] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,9,8,7,6,1] => [5,4,3,2,9,8,7,6,1] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [6,5,4,9,8,7,3,2,1] => [6,5,4,9,8,7,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [7,6,8,9,5,4,3,2,1] => [7,6,9,8,5,4,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [7,6,9,8,5,4,3,2,1] => [7,6,9,8,5,4,3,2,1] => ?
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [8,7,9,10,6,5,4,3,2,1] => [8,7,10,9,6,5,4,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [8,7,10,9,6,5,4,3,2,1] => [8,7,10,9,6,5,4,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,10,9,8,4,3,2,1] => [7,6,5,10,9,8,4,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,10,9,8,7,2,1] => [6,5,4,3,10,9,8,7,2,1] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,7,6,9,8,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,2,5,4,8,7,9,6,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [3,2,6,5,8,7,9,4,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [3,2,7,6,8,5,9,4,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,2,6,5,7,4,9,8,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,6,7,8,10,9,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St001280
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 1 = 2 - 1
[]
=> []
=> [] => ?
=> ? = 0 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 2 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1 = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 2 - 1
[]
=> []
=> [] => ? => ? = 0 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> ? => ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0,0]
=> ? => ? => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 2 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,7] => 1111000000 => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [4,3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [4,2,1,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [4,3,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [4,3,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,3,1,5,7,6] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,4,5,3,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,5,3,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,5,4,6,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000834
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [4,3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [4,2,1,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [4,3,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [4,3,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [6,5,4,3,7,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,3,1,5,7,6] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => [2,7,6,1,5,4,3] => [5,4,3,7,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,4,5,3,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,5,3,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => [2,7,6,5,1,4,3] => [4,3,7,6,5,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,5,4,6,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [2,7,6,5,4,1,3] => [3,7,6,5,4,1,2] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2 - 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000374
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 83%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [.,.]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [.,[[.,.],[[.,[[.,.],.]],.]]]
=> [2,5,6,4,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [.,[[.,.],[[[.,[.,.]],.],.]]]
=> [2,5,4,6,7,3,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [7,2,6,4,5,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [7,2,6,4,5,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [4,5,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> [4,6,5,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [3,4,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> [3,4,2,6,7,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [3,6,5,4,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> [3,5,6,4,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [4,3,5,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [4,3,6,5,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,5,3,4,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [7,5,3,4,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [3,4,5,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 83%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [.,.]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,3,2,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [3,6,5,4,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [3,5,6,4,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [3,4,5,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [3,5,4,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [.,[[.,.],[[.,[[.,.],.]],.]]]
=> [2,5,6,4,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => [7,6,3,5,4,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => [7,6,4,3,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [.,[[.,.],[[[.,[.,.]],.],.]]]
=> [2,5,4,6,7,3,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,6,3,4,5,2,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [7,2,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [7,2,6,4,5,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [7,2,6,4,5,3,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [4,5,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> [4,6,5,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [3,4,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> [3,4,2,6,7,5,1] => [7,3,2,6,5,4,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [3,6,5,4,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> [3,5,6,4,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [4,3,5,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [4,3,6,5,2,7,1] => [7,5,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,5,3,4,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [7,5,3,4,2,6,1] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [3,4,5,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 2 - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001333
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [7,5,4,3,2,6,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [8,7,5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,7,4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8,7,4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [8,7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [8,7,4,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [8,7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8,7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [8,7,3,5,6,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [8,7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [8,7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [8,7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8,7,3,4,5,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [8,6,5,4,3,7,2,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [8,6,4,3,5,7,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [8,6,3,5,4,7,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [8,5,3,4,7,6,2,1] => ([(0,1),(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [8,4,3,7,6,5,2,1] => ([(0,1),(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [8,3,7,6,5,4,2,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,7,5,6,4,2,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [8,4,3,7,5,6,2,1] => ([(0,1),(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [8,3,7,5,4,6,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [8,3,7,4,6,5,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [8,3,7,4,5,6,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [8,5,3,4,6,7,2,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [8,4,3,6,5,7,2,1] => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [8,3,6,5,4,7,2,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [8,3,6,4,5,7,2,1] => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [8,4,3,5,7,6,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [8,3,5,4,7,6,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [8,3,4,7,6,5,2,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [8,3,4,7,5,6,2,1] => ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [8,4,3,5,6,7,2,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> [8,3,5,4,6,7,2,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000912The number of maximal antichains in a poset. St000702The number of weak deficiencies of a permutation. St001665The number of pure excedances of a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000260The radius of a connected graph. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000779The tier of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000782The indicator function of whether a given perfect matching is an L & P matching. St001624The breadth of a lattice. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001773The number of minimal elements in Bruhat order not less than the signed 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 $eA$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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