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Your data matches 83 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 3
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 3
[4,1,2,3] => 4
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 4
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 87%●distinct values known / distinct values provided: 83%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 87%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[4,7,1,2,3,5,6,8] => ?
=> ? => ? = 4
[1,5,8,2,3,4,6,7] => ?
=> ? => ? = 1
[6,3,2,5,4,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 6
[2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 2
[6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 6
[2,1,8,5,4,7,6,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[2,1,4,3,10,7,6,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[7,3,2,8,9,10,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? = 7
[8,5,4,3,2,9,10,1,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? => ? = 8
[2,1,4,3,6,5,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,6,5,8,7,12,11,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,6,5,10,9,8,7,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,4,3,6,5,12,9,8,11,10,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,6,5,12,11,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,8,7,6,5,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,4,3,8,7,6,5,12,11,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,10,7,6,9,8,5,12,11] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,4,3,12,7,6,9,8,11,10,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,12,7,6,11,10,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,10,9,8,7,6,5,12,11] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,4,3,12,9,8,7,6,11,10,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,12,11,8,7,10,9,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,4,3,12,11,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,6,5,4,3,8,7,10,9,12,11] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,6,5,4,3,8,7,12,11,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,6,5,4,3,10,9,8,7,12,11] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,6,5,4,3,12,9,8,11,10,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,6,5,4,3,12,11,10,9,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,8,5,4,7,6,3,10,9,12,11] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,8,5,4,7,6,3,12,11,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,10,5,4,7,6,9,8,3,12,11] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,12,5,4,7,6,9,8,11,10,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,5,4,7,6,11,10,9,8,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,10,5,4,9,8,7,6,3,12,11] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,12,5,4,9,8,7,6,11,10,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,5,4,11,8,7,10,9,6,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,5,4,11,10,9,8,7,6,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,8,7,6,5,4,3,10,9,12,11] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,8,7,6,5,4,3,12,11,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? = 2
[2,1,10,7,6,5,4,9,8,3,12,11] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,12,7,6,5,4,9,8,11,10,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,7,6,5,4,11,10,9,8,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,10,9,6,5,8,7,4,3,12,11] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
[2,1,12,9,6,5,8,7,4,11,10,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,11,6,5,8,7,10,9,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,12,11,6,5,10,9,8,7,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 2
[2,1,10,9,8,7,6,5,4,3,12,11] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 84%●distinct values known / distinct values provided: 83%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 84%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1] => => ? = 1 - 1
[1,2] => [1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [2,1] => 1 => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 10 => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0 = 1 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 0 = 1 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 0 = 1 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 1 = 2 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 100 => 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0 = 1 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0010 => 0 = 1 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0100 => 0 = 1 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0100 => 0 = 1 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 0 = 1 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 0 = 1 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 0 = 1 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 0 = 1 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 0 = 1 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 0 = 1 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 0 = 1 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 0 = 1 - 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,5,6,8,7,4,3,1] => ? => ? = 2 - 1
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,6,4,3,1] => ? => ? = 2 - 1
[2,6,7,1,3,4,8,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ? => ? = 2 - 1
[2,1,6,3,4,7,8,5] => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ? => ? = 2 - 1
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,6,5,4,3,1,7,8] => ? => ? = 2 - 1
[3,4,6,8,1,2,5,7] => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,7,5,2,1] => ? => ? = 3 - 1
[3,4,7,8,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,4,7,8,6,5,2,1] => ? => ? = 3 - 1
[3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [3,5,4,6,2,1,8,7] => ? => ? = 3 - 1
[3,1,5,6,2,8,4,7] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,2,5,6,4,8,7,1] => ? => ? = 3 - 1
[3,6,7,1,2,8,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,5,4,8,2,1] => ? => ? = 3 - 1
[3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,6,5,7,8,4,2,1] => ? => ? = 3 - 1
[3,6,7,8,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,6,7,8,5,4,2,1] => ? => ? = 3 - 1
[4,1,5,2,7,8,3,6] => [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,2,7,8,6,1] => ? => ? = 4 - 1
[4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [4,5,7,6,3,8,2,1] => ? => ? = 4 - 1
[1,4,2,5,3,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,3,5,2,6,7,8] => ? => ? = 1 - 1
[4,7,1,2,3,5,6,8] => ?
=> ? => ? => ? = 4 - 1
[1,4,2,7,3,5,6,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,4,3,7,6,5,2,8] => ? => ? = 1 - 1
[1,5,2,6,3,8,4,7] => [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,4,6,3,8,7,2] => ? => ? = 1 - 1
[1,5,2,7,3,8,4,6] => [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,4,7,6,8,3,2] => ? => ? = 1 - 1
[1,5,2,8,3,4,6,7] => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,4,8,7,6,3,2] => ? => ? = 1 - 1
[1,5,8,2,3,4,6,7] => ?
=> ? => ? => ? = 1 - 1
[1,6,2,7,3,8,4,5] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,5,7,4,8,3,2] => ? => ? = 1 - 1
[1,6,2,3,4,7,5,8] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,6,5,4,3,7,2,8] => ? => ? = 1 - 1
[5,3,2,7,1,8,4,6] => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [5,4,3,7,6,8,2,1] => ? => ? = 5 - 1
[5,4,8,2,1,7,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,4,8,7,6,3,2,1] => ? => ? = 5 - 1
[3,5,1,8,2,7,6,4] => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,5,4,8,7,6,2,1] => ? => ? = 3 - 1
[3,6,1,8,7,2,5,4] => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,6,5,8,7,4,2,1] => ? => ? = 3 - 1
[6,3,2,5,4,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? = 6 - 1
[2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? = 2 - 1
[6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? = 6 - 1
[8,9,5,6,3,4,10,1,2,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
=> [8,9,7,6,5,4,10,3,2,1] => ? => ? = 8 - 1
[2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ? => ? = 2 - 1
[2,1,4,3,10,7,6,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? => ? = 2 - 1
[7,3,2,8,9,10,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? => ? = 7 - 1
[8,5,4,3,2,9,10,1,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? => ? => ? = 8 - 1
[2,1,4,3,6,5,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? => ? => ? = 2 - 1
[2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ? => ? = 2 - 1
[3,2,7,8,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,7,8,6,5,4,1] => ? => ? = 3 - 1
[3,6,7,4,5,8,1,2] => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,5,4,8,2,1] => ? => ? = 3 - 1
[2,6,7,4,5,1,8,3] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ? => ? = 2 - 1
[3,2,5,6,1,8,7,4] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,2,5,6,4,8,7,1] => ? => ? = 3 - 1
[2,6,7,3,1,5,8,4] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ? => ? = 2 - 1
[3,6,7,2,1,8,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,5,4,8,2,1] => ? => ? = 3 - 1
[3,2,5,4,8,1,6,7] => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,5,4,8,7,6,1] => ? => ? = 3 - 1
[5,6,1,2,3,8,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [5,6,4,3,2,8,7,1] => ? => ? = 5 - 1
[3,2,7,1,8,4,6,5] => [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,2,7,6,8,5,4,1] => ? => ? = 3 - 1
[5,1,8,2,7,6,4,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,4,8,7,6,3,2,1] => ? => ? = 5 - 1
[3,7,5,2,4,1,8,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,7,6,5,4,2,8,1] => ? => ? = 3 - 1
[2,4,5,7,3,1,8,6] => [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,6,3,8,1] => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1
[1,2] => [1,0,1,0]
=> [1,2] => [[1,2]]
=> 1
[2,1] => [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,5],[3],[4]]
=> 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,5,6,8,7,4,3,1] => ?
=> ? = 2
[2,5,1,7,3,4,8,6] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [2,5,4,7,6,3,8,1] => ?
=> ? = 2
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,6,4,3,1] => ?
=> ? = 2
[2,6,7,1,3,4,8,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ?
=> ? = 2
[2,1,6,3,4,7,8,5] => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ?
=> ? = 2
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,6,5,4,3,1,7,8] => ?
=> ? = 2
[3,4,6,8,1,2,5,7] => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,7,5,2,1] => ?
=> ? = 3
[3,4,7,8,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,4,7,8,6,5,2,1] => ?
=> ? = 3
[3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [3,5,4,6,2,1,8,7] => ?
=> ? = 3
[3,1,5,6,2,8,4,7] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,2,5,6,4,8,7,1] => ?
=> ? = 3
[1,3,5,6,2,7,4,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,4,7,2,8] => ?
=> ? = 1
[3,6,7,1,2,8,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,5,4,8,2,1] => ?
=> ? = 3
[3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,6,5,7,8,4,2,1] => ?
=> ? = 3
[3,6,7,8,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,6,7,8,5,4,2,1] => ?
=> ? = 3
[1,3,2,6,4,7,5,8] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,5,7,4,8] => ?
=> ? = 1
[4,1,5,6,8,2,3,7] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [4,3,5,6,8,7,2,1] => ?
=> ? = 4
[4,1,5,2,7,8,3,6] => [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,2,7,8,6,1] => ?
=> ? = 4
[4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [4,5,7,6,3,8,2,1] => ?
=> ? = 4
[4,5,1,7,8,2,3,6] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [4,5,3,7,8,6,2,1] => ?
=> ? = 4
[4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [4,5,7,8,6,3,2,1] => ?
=> ? = 4
[1,4,2,5,3,7,6,8] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ? = 1
[1,4,2,5,3,8,6,7] => [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3,5,2,8,7,6] => ?
=> ? = 1
[1,4,2,5,3,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,3,5,2,6,7,8] => ?
=> ? = 1
[4,6,7,8,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [4,6,7,8,5,3,2,1] => ?
=> ? = 4
[1,4,2,6,3,7,5,8] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ? = 1
[4,7,1,2,3,5,6,8] => ?
=> ? => ?
=> ? = 4
[1,4,2,7,3,5,6,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,4,3,7,6,5,2,8] => ?
=> ? = 1
[1,5,2,6,3,8,4,7] => [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,4,6,3,8,7,2] => ?
=> ? = 1
[1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,4,3,7,8] => ?
=> ? = 1
[1,5,2,7,3,8,4,6] => [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,4,7,6,8,3,2] => ?
=> ? = 1
[1,5,2,7,3,4,6,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,4,7,6,3,2,8] => ?
=> ? = 1
[1,5,2,8,3,4,6,7] => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,4,8,7,6,3,2] => ?
=> ? = 1
[1,5,8,2,3,4,6,7] => ?
=> ? => ?
=> ? = 1
[1,6,2,7,3,8,4,5] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,5,7,4,8,3,2] => ?
=> ? = 1
[1,6,2,3,4,7,5,8] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,6,5,4,3,7,2,8] => ?
=> ? = 1
[4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [4,7,6,5,3,8,2,1] => ?
=> ? = 4
[5,3,2,7,1,8,4,6] => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [5,4,3,7,6,8,2,1] => ?
=> ? = 5
[6,3,2,8,7,1,5,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,5,4,8,7,3,2,1] => ?
=> ? = 6
[5,4,8,2,1,7,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,4,8,7,6,3,2,1] => ?
=> ? = 5
[3,5,1,8,2,7,6,4] => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 3
[3,6,1,8,7,2,5,4] => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,6,5,8,7,4,2,1] => ?
=> ? = 3
[6,3,2,5,4,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ?
=> ? = 6
[2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ?
=> ? = 2
[6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ?
=> ? = 6
[2,1,8,5,4,7,6,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,1,8,7,6,5,4,3,10,9] => [[1,3,9],[2,4,10],[5],[6],[7],[8]]
=> ? = 2
[8,9,5,6,3,4,10,1,2,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
=> [8,9,7,6,5,4,10,3,2,1] => ?
=> ? = 8
[2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ?
=> ? = 2
[2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,1,8,7,6,5,4,3,10,9] => [[1,3,9],[2,4,10],[5],[6],[7],[8]]
=> ? = 2
[2,1,4,3,10,7,6,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? => ?
=> ? = 2
[7,3,2,8,9,10,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 7
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000010
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 92%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => [1,0]
=> []
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 3 - 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2 = 3 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3 = 4 - 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3 = 4 - 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 7 - 1
[7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 7 - 1
[7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 7 - 1
[7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 7 - 1
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? = 8 - 1
[7,6,8,5,4,3,2,1] => [2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 7 - 1
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2]
=> ? = 7 - 1
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2]
=> ? = 7 - 1
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,1,1]
=> ? = 8 - 1
[7,6,5,4,8,3,2,1] => [2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 7 - 1
[6,5,7,4,8,3,2,1] => [3,4,2,5,1,6,7,8] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 6 - 1
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2]
=> ? = 7 - 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,1,1]
=> ? = 8 - 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ? = 8 - 1
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3]
=> ? = 6 - 1
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2]
=> ? = 7 - 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ? = 8 - 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,2,2,1]
=> ? = 8 - 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ? = 8 - 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2,1,1]
=> ? = 8 - 1
[6,7,8,5,2,3,4,1] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3]
=> ? = 6 - 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ? = 8 - 1
[7,8,3,2,4,5,6,1] => [2,1,6,7,5,4,3,8] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,2]
=> ? = 7 - 1
[7,6,5,4,3,2,8,1] => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 7 - 1
[6,5,7,4,3,2,8,1] => [3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1]
=> ? = 6 - 1
[6,5,4,3,7,2,8,1] => [3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ? = 6 - 1
[8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 8 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> ? = 7 - 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,1,1]
=> ? = 8 - 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2,1]
=> ? = 8 - 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2,1]
=> ? = 8 - 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2,1]
=> ? = 8 - 1
[8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2,1]
=> ? = 8 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 7 - 1
[5,6,7,8,3,4,1,2] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4]
=> ? = 5 - 1
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,6,2,2,2,2]
=> ? = 7 - 1
[8,7,6,5,4,2,1,3] => [1,2,3,4,5,7,8,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> ? = 8 - 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,1]
=> ? = 7 - 1
[8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> ? = 8 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 6 - 1
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,1,1,1]
=> ? = 8 - 1
[8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ? = 8 - 1
[8,6,7,4,5,1,2,3] => [1,3,2,5,4,8,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 8 - 1
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,2,1]
=> ? = 8 - 1
[7,8,4,5,6,1,2,3] => [2,1,5,4,3,8,7,6] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,2]
=> ? = 7 - 1
[6,7,4,5,8,1,2,3] => [3,2,5,4,1,8,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2]
=> ? = 6 - 1
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,1]
=> ? = 7 - 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1,1]
=> ? = 7 - 1
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2]
=> ? = 6 - 1
Description
The length of the partition.
Matching statistic: St000439
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 92%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ? = 6 + 1
[6,5,4,3,7,2,8,1] => [6,5,4,3,8,2,7,1] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[4,3,2,5,6,7,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[4,2,3,5,6,7,8,1] => [4,2,8,7,6,5,3,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[3,2,4,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ? = 8 + 1
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ? = 8 + 1
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ? = 8 + 1
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => ?
=> ? = 7 + 1
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[4,3,6,5,8,7,2,1] => [4,3,8,7,6,5,2,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[3,2,6,5,4,8,7,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[4,3,6,5,7,2,8,1] => [4,3,8,7,6,2,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[3,2,6,5,7,4,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[4,3,5,2,7,6,8,1] => [4,3,8,2,7,6,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[3,2,5,4,7,6,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,4,1,6,8,7,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,4,1,7,6,8,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[3,2,5,4,1,8,7,6] => [3,2,8,7,1,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,4,5,8,7,6] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[4,3,5,2,6,1,8,7] => [4,3,8,2,7,1,6,5] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[3,2,5,4,6,1,8,7] => [3,2,8,7,6,1,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,4,1,6,5,8,7] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,7,6,5,4,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[5,4,3,2,7,6,1,8] => [5,4,3,2,8,7,1,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 5 + 1
[3,2,5,4,7,6,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[4,2,3,1,8,6,7,5] => [4,2,8,1,7,6,5,3] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,3,2,7,5,6,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[4,3,2,8,5,7,6,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[3,1,4,2,6,5,8,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,4,2,6,8,5,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,4,2,7,5,8,6] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,5,2,6,4,8,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,5,6,2,8,4,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,5,2,7,4,8,6] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,6,2,7,4,8,5] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[3,1,2,6,7,4,8,5] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,3,2,6,4,5,7,8] => ? => ?
=> ? = 1 + 1
[1,3,2,7,4,5,6,8] => ? => ?
=> ? = 1 + 1
[3,1,2,4,5,8,6,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[4,1,5,2,6,3,8,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,1,5,6,8,2,3,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,1,5,2,7,3,8,6] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,1,5,2,7,8,3,6] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,1,6,2,7,3,8,5] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[4,1,2,3,6,5,8,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ?
=> ? = 6
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ?
=> ? = 8
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ?
=> ? = 8
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ?
=> ? = 8
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => ?
=> ?
=> ? = 7
[7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,6,4,5,3,2,1,8] => [7,6,4,8,3,2,1,5] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,6,4,3,5,2,1,8] => [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7
[7,6,3,4,5,2,1,8] => [7,6,3,8,5,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,5,4,3,6,2,1,8] => [7,5,4,3,8,2,1,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7
[7,4,3,6,5,2,1,8] => [7,4,3,8,6,2,1,5] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,6,3,5,4,2,1,8] => [7,6,3,8,5,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,3,2,6,4,5,7,8] => ? => ?
=> ?
=> ? = 1
[1,3,2,7,4,5,6,8] => ? => ?
=> ?
=> ? = 1
[1,4,2,3,5,7,6,8] => ? => ?
=> ?
=> ? = 1
[1,6,2,7,3,4,5,8] => ? => ?
=> ?
=> ? = 1
[1,6,2,8,3,4,5,7] => ? => ?
=> ?
=> ? = 1
[1,6,2,3,4,8,5,7] => ? => ?
=> ?
=> ? = 1
[7,3,2,5,4,8,1,6] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 7
[7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7
[7,3,2,6,8,4,1,5] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,3,2,8,6,5,1,4] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,10,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[6,3,2,5,4,1,8,7,10,9] => [6,3,2,10,9,1,8,7,5,4] => ?
=> ?
=> ? = 6
[8,3,2,5,4,7,6,1,10,9] => [8,3,2,10,9,7,6,1,5,4] => ?
=> ?
=> ? = 8
[10,3,2,5,4,7,6,9,8,1] => [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,1,6,5,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,10,9,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[8,5,4,3,2,7,6,1,10,9] => [8,5,4,3,2,10,9,1,7,6] => ?
=> ?
=> ? = 8
[10,5,4,3,2,7,6,9,8,1] => [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,1,8,5,4,7,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[8,7,4,3,6,5,2,1,10,9] => [8,7,4,3,10,9,2,1,6,5] => ?
=> ?
=> ? = 8
[10,7,4,3,6,5,2,9,8,1] => [10,7,4,3,9,8,2,6,5,1] => ?
=> ?
=> ? = 10
[2,1,10,5,4,7,6,9,8,3] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[10,9,4,3,6,5,8,7,2,1] => [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[8,9,5,6,3,4,10,1,2,7] => [8,10,5,9,3,7,6,1,4,2] => ?
=> ?
=> ? = 8
[2,1,4,3,8,7,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,10,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[8,3,2,7,6,5,4,1,10,9] => [8,3,2,10,9,7,6,1,5,4] => ?
=> ?
=> ? = 8
[10,3,2,7,6,5,4,9,8,1] => [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,1,8,7,6,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,10,1,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 9
[10,7,6,5,4,3,2,9,8,1] => [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ?
=> ? = 6
[6,5,4,3,7,2,8,1] => [6,5,4,3,8,2,7,1] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[5,4,6,3,7,2,8,1] => [5,4,8,3,7,2,6,1] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 5
[5,2,3,4,6,7,8,1] => [5,2,8,7,6,4,3,1] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 5
[4,3,2,5,6,7,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[4,2,3,5,6,7,8,1] => [4,2,8,7,6,5,3,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,2,4,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ?
=> ? = 8
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ?
=> ? = 8
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ?
=> ? = 8
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => ?
=> ?
=> ? = 7
[6,5,4,7,3,2,1,8] => [6,5,4,8,3,2,1,7] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6
[5,1,2,3,4,6,7,8] => [5,1,8,7,6,4,3,2] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 5
[4,3,6,5,8,7,2,1] => [4,3,8,7,6,5,2,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,2,6,5,4,8,7,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[4,3,6,5,7,2,8,1] => [4,3,8,7,6,2,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,2,6,5,7,4,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[4,3,5,2,7,6,8,1] => [4,3,8,2,7,6,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,2,5,4,7,6,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,2,4,1,6,8,7,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,2,4,1,7,6,8,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,2,5,4,1,8,7,6] => [3,2,8,7,1,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[4,3,5,2,6,1,8,7] => [4,3,8,2,7,1,6,5] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,2,5,4,6,1,8,7] => [3,2,8,7,6,1,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,2,4,1,6,5,8,7] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,2,7,6,5,4,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[5,4,3,2,7,6,1,8] => [5,4,3,2,8,7,1,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[3,2,5,4,7,6,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[4,2,3,1,8,6,7,5] => [4,2,8,1,7,6,5,3] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,3,2,7,5,6,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[4,3,2,8,5,7,6,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6
[3,1,4,2,6,5,8,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,4,2,6,8,5,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,4,2,7,5,8,6] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,5,2,6,4,8,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,5,6,2,8,4,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,5,2,7,4,8,6] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,6,2,7,4,8,5] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,1,2,6,7,4,8,5] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,3,2,6,4,5,7,8] => ? => ?
=> ?
=> ? = 1
[1,3,2,7,4,5,6,8] => ? => ?
=> ?
=> ? = 1
[3,1,2,4,5,8,6,7] => [3,1,8,7,6,5,4,2] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[4,1,5,2,6,3,8,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,1,5,6,8,2,3,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,1,5,2,7,3,8,6] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,1,5,2,7,8,3,6] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,1,6,2,7,3,8,5] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,1,2,3,6,5,8,7] => [4,1,8,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[1,4,2,3,5,7,6,8] => ? => ?
=> ?
=> ? = 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000971
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 7
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ?
=> ? = 6
[6,5,4,3,7,2,8,1] => [6,5,4,3,8,2,7,1] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> {{1,2,7,8},{3,4,5,6}}
=> ? = 6
[4,3,2,5,6,7,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 4
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ?
=> ? = 8
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ?
=> ? = 8
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ?
=> ? = 8
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => ?
=> ?
=> ? = 7
[7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[7,6,4,5,3,2,1,8] => [7,6,4,8,3,2,1,5] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[6,5,4,7,3,2,1,8] => [6,5,4,8,3,2,1,7] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 6
[7,6,4,3,5,2,1,8] => [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 7
[7,6,3,4,5,2,1,8] => [7,6,3,8,5,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[7,5,4,3,6,2,1,8] => [7,5,4,3,8,2,1,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 7
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 5
[4,3,2,1,5,6,7,8] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 3
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 3
[3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 3
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 5
[3,2,1,4,5,8,7,6] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 3
[4,3,2,1,6,5,8,7] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[5,4,3,2,7,6,1,8] => [5,4,3,2,8,7,1,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 5
[4,3,2,7,5,6,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 4
[4,3,2,8,5,7,6,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 4
[6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 6
[7,4,3,6,5,2,1,8] => [7,4,3,8,6,2,1,5] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[7,6,3,5,4,2,1,8] => [7,6,3,8,5,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[1,3,2,6,4,5,7,8] => ? => ?
=> ?
=> ? = 1
[1,3,2,7,4,5,6,8] => ? => ?
=> ?
=> ? = 1
[1,4,2,3,5,7,6,8] => ? => ?
=> ?
=> ? = 1
[1,6,2,7,3,4,5,8] => ? => ?
=> ?
=> ? = 1
[1,6,2,8,3,4,5,7] => ? => ?
=> ?
=> ? = 1
[1,6,2,3,4,8,5,7] => ? => ?
=> ?
=> ? = 1
[5,3,2,6,1,4,8,7] => [5,3,2,8,1,7,6,4] => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 5
[7,3,2,5,4,8,1,6] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 7
[7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 7
[7,3,2,6,8,4,1,5] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[6,3,2,7,8,1,4,5] => [6,3,2,8,7,1,5,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 6
[5,3,2,7,1,8,4,6] => [5,3,2,8,1,7,6,4] => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 5
[4,3,2,1,7,8,5,6] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[5,3,2,8,1,7,6,4] => [5,3,2,8,1,7,6,4] => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 5
[6,3,2,8,7,1,5,4] => [6,3,2,8,7,1,5,4] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 6
[7,3,2,8,6,5,1,4] => [7,3,2,8,6,5,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 7
[2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> ? = 2
[4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,10,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9,10}}
=> ? = 4
[6,3,2,5,4,1,8,7,10,9] => [6,3,2,10,9,1,8,7,5,4] => ?
=> ?
=> ? = 6
[8,3,2,5,4,7,6,1,10,9] => [8,3,2,10,9,7,6,1,5,4] => ?
=> ?
=> ? = 8
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001733
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[7,8,5,6,4,3,2,1] => [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,8,6,4,5,3,2,1] => [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,8,6,5,3,4,2,1] => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[6,7,8,3,4,5,2,1] => [6,8,7,3,5,4,2,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[7,8,6,5,4,2,3,1] => [7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => ?
=> ?
=> ? = 6
[7,8,3,2,4,5,6,1] => [7,8,3,2,6,5,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[3,4,5,2,6,7,8,1] => [3,8,7,2,6,5,4,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,2,5,6,7,8,1] => [3,8,2,7,6,5,4,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[7,8,6,5,4,3,1,2] => [7,8,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[7,8,3,4,5,6,1,2] => [7,8,3,6,5,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[3,4,5,6,7,8,1,2] => [3,8,7,6,5,4,1,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[6,7,8,5,4,1,2,3] => [6,8,7,5,4,1,3,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ?
=> ? = 8
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ?
=> ? = 8
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ?
=> ? = 8
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => ?
=> ?
=> ? = 7
[6,7,4,5,8,1,2,3] => [6,8,4,7,5,1,3,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[7,8,5,6,1,2,3,4] => [7,8,5,6,1,4,3,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[6,7,8,1,2,3,4,5] => [6,8,7,1,5,4,3,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[7,8,5,4,2,1,3,6] => [7,8,5,4,2,1,6,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,8,5,4,1,2,3,6] => [7,8,5,4,1,6,3,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,8,1,2,3,4,5,6] => [7,8,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[6,7,5,4,3,2,1,8] => [6,8,5,4,3,2,1,7] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[5,6,7,4,3,2,1,8] => [5,8,7,4,3,2,1,6] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[6,7,5,4,2,3,1,8] => [6,8,5,4,2,7,1,3] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[5,6,7,2,3,4,1,8] => [5,8,7,2,6,4,1,3] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[5,6,7,1,2,3,4,8] => [5,8,7,1,6,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[6,7,1,2,3,4,5,8] => [6,8,1,7,5,4,3,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[4,5,6,1,2,3,7,8] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[3,4,1,2,5,6,7,8] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,5,8,7,6,4,2,1] => [3,8,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,7,6,4,3,2,1,8] => [5,8,7,4,3,2,1,6] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[4,7,6,5,3,2,1,8] => [4,8,7,6,3,2,1,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,6,8,1,2,5,7] => [3,8,7,6,1,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,7,8,1,2,5,6] => [3,8,7,6,1,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,5,6,8,1,2,4,7] => [3,8,7,6,1,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,5,7,8,1,2,4,6] => [3,8,7,6,1,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,6,7,1,2,8,4,5] => [3,8,7,1,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000383The last part of an integer composition. St000759The smallest missing part in an integer partition. St000007The number of saliances of the permutation. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000839The largest opener of a set partition. St000069The number of maximal elements of a poset. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000675The number of centered multitunnels of a Dyck path. St000068The number of minimal elements in a poset. St000738The first entry in the last row of a standard tableau. St000505The biggest entry in the block containing the 1. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000234The number of global ascents of a permutation. St000740The last entry of a permutation. St000702The number of weak deficiencies of a permutation. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St000989The number of final rises of a permutation. St000203The number of external nodes of a binary tree. St000653The last descent of a permutation. St000501The size of the first part in the decomposition of a permutation. St001497The position of the largest weak excedence of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000843The decomposition number of a perfect matching. 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. St000990The first ascent of a permutation. St000734The last entry in the first row of a standard tableau. St000051The size of the left subtree of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. 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. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St001480The number of simple summands of the module J^2/J^3. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001434The number of negative sum pairs of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
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