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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => 0
[1,2] => [1] => 0
[2,1] => [1] => 0
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 2
[1,2,4,3] => [1,2,3] => 2
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 2
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 2
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 3
[1,2,3,5,4] => [1,2,3,4] => 3
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 3
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000010
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> 0
[2,1] => [1] => [1]
=> []
=> 0
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 1
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 1
[2,1,3] => [2,1] => [2]
=> []
=> 0
[2,3,1] => [2,1] => [2]
=> []
=> 0
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 1
[3,2,1] => [2,1] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> 1
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> 1
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1,4] => [3,2,1] => [3]
=> []
=> 0
[3,2,4,1] => [3,2,1] => [3]
=> []
=> 0
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> 1
[3,4,2,1] => [3,2,1] => [3]
=> []
=> 0
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 1
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 1
[4,3,2,1] => [3,2,1] => [3]
=> []
=> 0
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> 1
Description
The length of the partition.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => []
=> []
=> 0
[1,2] => [1] => [1]
=> [[1]]
=> 0
[2,1] => [1] => [1]
=> [[1]]
=> 0
[1,2,3] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[1,3,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[2,1,3] => [2,1] => [2]
=> [[1,2]]
=> 0
[2,3,1] => [2,1] => [2]
=> [[1,2]]
=> 0
[3,1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[3,2,1] => [2,1] => [2]
=> [[1,2]]
=> 0
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,3,2,4] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,4,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[2,3,1,4] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 1
[2,3,4,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[2,4,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 1
[3,1,2,4] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 1
[3,1,4,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 1
[3,2,1,4] => [3,2,1] => [3]
=> [[1,2,3]]
=> 0
[3,2,4,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 0
[3,4,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 1
[3,4,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 0
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[4,1,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[4,2,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 1
[4,3,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 1
[4,3,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 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: St000288
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => []
=> => ? = 0 + 1
[1,2] => [1] => [1]
=> 10 => 1 = 0 + 1
[2,1] => [1] => [1]
=> 10 => 1 = 0 + 1
[1,2,3] => [1,2] => [1,1]
=> 110 => 2 = 1 + 1
[1,3,2] => [1,2] => [1,1]
=> 110 => 2 = 1 + 1
[2,1,3] => [2,1] => [2]
=> 100 => 1 = 0 + 1
[2,3,1] => [2,1] => [2]
=> 100 => 1 = 0 + 1
[3,1,2] => [1,2] => [1,1]
=> 110 => 2 = 1 + 1
[3,2,1] => [2,1] => [2]
=> 100 => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> 1010 => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> 1010 => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> 1010 => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1]
=> 1010 => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [2,1]
=> 1010 => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1]
=> 1010 => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [2,1]
=> 1010 => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,1]
=> 1010 => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3]
=> 1000 => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [3]
=> 1000 => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [2,1]
=> 1010 => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3]
=> 1000 => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> 1110 => 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> 1010 => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1]
=> 1010 => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [2,1]
=> 1010 => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3]
=> 1000 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> 11110 => 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> 11110 => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> 11110 => 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> 10010 => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> 10010 => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> 10110 => 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> 10010 => 2 = 1 + 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [10,1]
=> 100000000010 => ? = 1 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000733
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => []
=> []
=> ? = 0 + 1
[1,2] => [1] => [1]
=> [[1]]
=> 1 = 0 + 1
[2,1] => [1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,3,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[3,1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[3,2,1] => [2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2 = 1 + 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000093
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Values
[1] => [] => [] => ?
=> ? = 0 + 1
[1,2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[2,1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2,3] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[1,3,2] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3,1,2] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [1,1,3,4] => ([(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 9 + 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [1,9] => ([(8,9)],10)
=> ? = 8 + 1
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ([(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)
=> ? = 1 + 1
[10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 9 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [1,9] => ([(8,9)],10)
=> ? = 8 + 1
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Values
[1] => [] => [] => ?
=> ? = 0 + 1
[1,2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[2,1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2,3] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[1,3,2] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3,1,2] => [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [1,1,3,4] => ([(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 9 + 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [1,9] => ([(8,9)],10)
=> ? = 8 + 1
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ([(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)
=> ? = 1 + 1
[10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 8 + 1
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 7 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 9 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [1,9] => ([(8,9)],10)
=> ? = 8 + 1
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000676
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 97%●distinct values known / distinct values provided: 80%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 97%●distinct values known / distinct values provided: 80%
Values
[1] => [] => .
=> ?
=> ? = 0 + 1
[1,2] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7 + 1
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5 + 1
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7 + 1
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6 + 1
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7 + 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6 + 1
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6 + 1
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [[[[.,.],.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 5 + 1
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,.]]]
=> ?
=> ? = 5 + 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 + 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 + 1
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[9,10,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[9,8,10,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000024
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Values
[1] => [] => .
=> ?
=> ? = 0
[1,2] => [1] => [.,.]
=> [1,0]
=> 0
[2,1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[2,3,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 5
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 1
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [[[[.,.],.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 5
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,.]]]
=> ?
=> ? = 5
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 9
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 6
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 97%●distinct values known / distinct values provided: 70%
Values
[1] => [] => .
=> ?
=> ? = 0
[1,2] => [1] => [.,.]
=> [1,0]
=> 0
[2,1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[2,3,4,5,6,8,9,1,7] => [2,3,4,5,6,8,1,7] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[2,3,4,5,7,8,9,1,6] => [2,3,4,5,7,8,1,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[2,3,4,5,6,1,9,7,8] => [2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
[2,3,4,5,6,7,9,10,1,8] => [2,3,4,5,6,7,9,1,8] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 1
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [[[.,[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 6
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [[[[.,.],.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 5
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,.]]]
=> ?
=> ? = 5
[1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
Description
The number of valleys of the Dyck path.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000167The number of leaves of an ordered tree. St000702The number of weak deficiencies of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. 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. St000021The number of descents of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001427The number of descents of a signed permutation. St001863The number of weak excedances of a signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function.
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