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Matching statistic: St000157
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St001214
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,1,1]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 1
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St001389
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000292
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 001 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0
Description
The number of ascents of a binary word.
Matching statistic: St000389
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 001 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 001 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000919
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [5,3,1,2,4,6,7,8] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [1,2,4,5,6,7,8,9,3] => [.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2,10] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [1,2,3,5,6,7,8,9,4] => [.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [1,2,4,5,6,7,8,9,10,3] => [.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,10,1,2] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,8,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8,7,6,4,5,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [1,2,3,4,5,6,8,9,7] => [.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,9,6,5,4,3,2,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,10,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,8,10,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,10,9] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [8,6,7,9,5,4,3,2,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,8,7,10,6,5,4,3,2,1] => [1,2,3,4,5,6,7,10,8,9] => [.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,10,5,4,3,2,1] => [1,2,3,4,5,6,10,7,8,9] => [.,[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,10,4,3,2,1] => [1,2,3,4,5,10,6,7,8,9] => [.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,10,3,2,1] => [1,2,3,4,10,5,6,7,8,9] => [.,[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,10,2,1] => [1,2,3,10,4,5,6,7,8,9] => [.,[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,2,10,1] => [1,2,10,3,4,5,6,7,8,9] => [.,[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [6,7,8,9,10,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [9,7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [8,9,6,7,4,5,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000568
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [5,3,1,2,4,6,7,8] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [1,2,4,5,6,7,8,9,3] => [.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2,10] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [1,2,3,5,6,7,8,9,4] => [.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [1,2,4,5,6,7,8,9,10,3] => [.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,10,1,2] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,8,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8,7,6,4,5,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ? = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [1,2,3,4,5,6,8,9,7] => [.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,9,6,5,4,3,2,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,10,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,8,10,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,10,9] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [8,6,7,9,5,4,3,2,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,8,7,10,6,5,4,3,2,1] => [1,2,3,4,5,6,7,10,8,9] => [.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,10,5,4,3,2,1] => [1,2,3,4,5,6,10,7,8,9] => [.,[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,10,4,3,2,1] => [1,2,3,4,5,10,6,7,8,9] => [.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,10,3,2,1] => [1,2,3,4,10,5,6,7,8,9] => [.,[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,10,2,1] => [1,2,3,10,4,5,6,7,8,9] => [.,[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,2,10,1] => [1,2,10,3,4,5,6,7,8,9] => [.,[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [6,7,8,9,10,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [9,7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [8,9,6,7,4,5,1,2,3] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St000884
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [1,7,2,3,4,6,5] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [1,7,2,4,3,5,6] => ? = 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [1,7,2,4,3,5,6] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,1,7] => [1,7,2,5,3,4,6] => ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [1,7,2,6,3,5,4] => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [1,7,2,3,5,4,6] => ? = 2
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [1,7,2,3,6,4,5] => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [1,7,2,3,6,4,5] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,1,7] => [1,7,2,4,5,3,6] => ? = 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [1,7,2,4,6,3,5] => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [1,7,2,5,6,3,4] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [1,7,2,5,6,3,4] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [1,7,2,3,4,6,5] => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [1,7,2,3,5,6,4] => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,7,2,4,5,6,3] => ? = 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St001469The holeyness of a permutation. St000354The number of recoils of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001665The number of pure excedances of a permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000353The number of inner valleys of a permutation. St000632The jump number of the poset. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000092The number of outer peaks of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001358The largest degree of a regular subgraph of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001427The number of descents of a signed permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St000454The largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000058The order of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000862The number of parts of the shifted shape of a permutation. St000805The number of peaks of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. St001823The Stasinski-Voll length of a signed permutation. St001871The number of triconnected components of a graph. St001905The number of preferred parking spots in a parking function less than the index of the car. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001948The number of augmented double ascents of a permutation. 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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