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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St000759
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 2
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 3
[1,1,1]
=> 2
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 3
[1,1,1,1]
=> 2
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 3
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 2
[6]
=> 1
[5,1]
=> 2
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 1
[3,2,1]
=> 4
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 3
[1,1,1,1,1,1]
=> 2
[7]
=> 1
[6,1]
=> 2
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 3
[4,1,1,1]
=> 2
[3,3,1]
=> 2
[3,2,2]
=> 1
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 3
[1,1,1,1,1,1,1]
=> 2
[8]
=> 1
[7,1]
=> 2
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 3
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0 => 2
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 11110 => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0110 => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 111110 => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 01110 => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1010 => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 00111 => 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 011111 => 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 1111110 => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 011110 => 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 10110 => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 01110 => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 01011 => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 00111 => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 001111 => 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 0111111 => 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 11111110 => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => 0111110 => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 101110 => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 011110 => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 11010 => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 00110 => 3
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,8,7,6,5,4,3,2,1,11] => ? => ? = 2
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,8,6,5,4,3,2,1,10] => ? => ? = 2
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ? => ? = 2
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,6,5,4,2,1,8] => ? => ? = 2
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,7,6,9,5,4,3,2] => ? => ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ? => ? = 4
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,9,8,7,6,5,4,3,2] => ? => ? = 3
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ? => ? = 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,9,8,7,6,5,4,3,2,1,12] => ? => ? = 2
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,9,7,6,5,4,3,2,1,11] => ? => ? = 2
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,8,7,5,4,3,2,1,10] => ? => ? = 2
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,6,5,4,8,3,2,1,9] => ? => ? = 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,6,4,3,2,1,9] => ? => ? = 3
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,6,7,5,4,2,1,8] => ? => ? = 3
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,6,7,5,4,8,3,2] => ? => ? = 3
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,8,7,6,5,9,4,3,2] => ? => ? = 2
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,7,8,6,9,5,4,3,2] => ? => ? = 3
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9,8,7,10,6,5,4,3,2] => ? => ? = 2
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,7,4,3,2] => ? => ? = 4
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,6,8,9,7,5,4,3,2] => ? => ? = 4
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,11,8,7,6,5,4,3,2] => ? => ? = 2
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,5,9,8,7,6,4,3,2] => ? => ? = 3
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9,11,10,8,7,6,5,4,3,2] => 0011111111 => ? = 3
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,12,10,9,8,7,6,5,4,3,2] => ? => ? = 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 2
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 2
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [6,5,4,7,3,8,2,1,9] => ? => ? = 1
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[]
=> []
=> [] => ? => ? = 1
[11,7,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,6,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[10,6,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[11,7,5,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 2
[11,9,7,5,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2
[9,7,5,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2
[11,8,7,5,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[9,6,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,5,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,5,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,6,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[11,8,6,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[11,8,6,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[12,9,7,5,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2] => 2
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,2] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,1,1,2] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,1,1,1,2] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [2,1,2] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [3,1,1] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,1,1,1] => 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [2,1,1,2] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,2,2] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,1,2] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,2,1] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,2,1] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,1,1] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1] => 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [3,1,1,1] => 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [2,1,1,1,2] => 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [1,2,1,2] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [2,1,1,2] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,2] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [2,2,1,1] => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [3,1,1,1] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [3,1,1,1,1] => 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [2,1,1,1,1,2] => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [1,2,1,1,2] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [2,1,1,1,2] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,1,2,2] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [3,1,2] => 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1,1] => ? = 2
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,8,7,6,5,4,3,2,1,11] => ? => ? = 2
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,8,6,5,4,3,2,1,10] => ? => ? = 2
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,5,4,3,2,1,9] => ? => ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ? => ? = 2
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,7,6,9,5,4,3,2] => ? => ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,9,8,7,6,5,4,3,2] => [3,1,1,1,1,1,1,1,1] => ? = 3
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ? => ? = 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,9,8,7,6,5,4,3,2,1,12] => ? => ? = 2
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,9,7,6,5,4,3,2,1,11] => ? => ? = 2
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,9,6,5,4,3,2,1,10] => ? => ? = 3
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,8,7,5,4,3,2,1,10] => ? => ? = 2
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,6,5,4,8,3,2,1,9] => ? => ? = 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [6,7,5,8,4,3,2,1,9] => ? => ? = 2
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,6,4,3,2,1,9] => ? => ? = 3
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,6,7,5,4,2,1,8] => ? => ? = 3
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,6,7,5,4,8,3,2] => ? => ? = 3
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,8,7,6,5,9,4,3,2] => ? => ? = 2
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,7,8,6,9,5,4,3,2] => ? => ? = 3
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9,8,7,10,6,5,4,3,2] => ? => ? = 2
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,6,8,9,7,5,4,3,2] => ? => ? = 4
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,11,8,7,6,5,4,3,2] => ? => ? = 2
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,5,9,8,7,6,4,3,2] => ? => ? = 3
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9,11,10,8,7,6,5,4,3,2] => [3,1,1,1,1,1,1,1,1] => ? = 3
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,12,10,9,8,7,6,5,4,3,2] => ? => ? = 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 2
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [5,6,4,3,2,7,1,8] => ? => ? = 2
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 2
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [6,5,4,7,3,8,2,1,9] => ? => ? = 1
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[]
=> []
=> [] => [] => ? = 1
[11,7,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,6,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[10,6,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[11,7,5,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 2
[11,9,7,5,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2
[9,7,5,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ? => ? = 2
[11,8,7,5,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[9,6,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,5,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,5,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[9,6,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[10,7,6,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
[11,8,6,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3 + 1
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 2 + 1
[8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ?
=> ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 3
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 3
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,1,0,0,0]
=> ?
=> ? = 3
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,1,0]
=> ?
=> ? = 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 3
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001050
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,6,7},{5}}
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2,3,4,7},{5,6}}
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> {{1},{2,3,6},{4},{5}}
=> 3
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,11},{10}}
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9,10},{8}}
=> ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8,9}}
=> ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8,9},{6}}
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7},{8}}
=> ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,9},{6,7,8}}
=> ? = 2
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5,6}}
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,7},{6}}
=> ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> {{1},{2,3,8},{4,5,6,7}}
=> ? = 2
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 2
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,7,8},{5},{9}}
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,6},{5},{8}}
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,8},{6},{7},{9}}
=> ? = 4
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8,9},{7},{10}}
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,5,6},{8}}
=> ? = 3
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,8},{5,6,7},{9}}
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,9},{7,8},{10}}
=> ? = 3
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,10},{9},{11}}
=> ? = 3
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9,10,11},{12}}
=> ? = 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,10,11},{9}}
=> ? = 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 2
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9,10},{7}}
=> ? = 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8},{9}}
=> ? = 3
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,10},{7,8,9}}
=> ? = 2
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8,9},{5}}
=> ? = 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,9},{6},{8}}
=> ? = 2
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,8,9},{6,7}}
=> ? = 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,9},{6,8},{7}}
=> ? = 3
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,9},{5,6,7,8}}
=> ? = 2
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,5,6,8},{4},{7}}
=> ? = 2
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6,7}}
=> ? = 2
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6},{7}}
=> ? = 3
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> {{1},{2,3,8},{4,5,7},{6}}
=> ? = 3
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> {{1},{2,8},{3,4,5,6,7}}
=> ? = 2
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 2
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,4,5,7},{3},{6},{8}}
=> ? = 3
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,5,6,7,8},{4},{9}}
=> ? = 2
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4},{5},{8}}
=> ? = 2
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4},{5,6},{8}}
=> ? = 3
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,8},{5},{7},{9}}
=> ? = 3
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8,9},{6},{10}}
=> ? = 2
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,5},{6},{8}}
=> ? = 4
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,7,8},{5,6},{9}}
=> ? = 2
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000971
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 90%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}}
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}}
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}}
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => {{1,2},{3},{4},{5},{6},{7}}
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => {{1,6,7},{2},{3},{4},{5}}
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => {{1},{2,5,6},{3},{4}}
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => {{1,4},{2},{3},{5,6}}
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => {{1,2},{3,6},{4},{5}}
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => {{1,2,5},{3},{4},{6}}
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => {{1,2,7},{3},{4},{5},{6}}
=> 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => {{1},{2,6,7},{3},{4},{5}}
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => {{1,5},{2},{3},{4},{6,7}}
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => {{1},{2},{3,5,6},{4}}
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => {{1,4},{2,5,6},{3}}
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => {{1,3},{2},{4},{5,6}}
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}}
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => {{1},{2,7,8},{3},{4},{5},{6}}
=> ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> ? = 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => {{1},{2,8,9},{3},{4},{5},{6},{7}}
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => {{1,7},{2},{3},{4},{5},{6},{8,9}}
=> ? = 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,2,7,3,4,5,8,6] => {{1},{2},{3,7,8},{4},{5},{6}}
=> ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => {{1,6},{2,7,8},{3},{4},{5}}
=> ? = 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => {{1,5},{2},{3},{4},{6},{7,8}}
=> ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => {{1,2},{3},{4,8},{5},{6},{7}}
=> ? = 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => {{1,2,7},{3,8},{4},{5},{6}}
=> ? = 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => {{1,2},{3,9},{4},{5},{6},{7},{8}}
=> ? = 2
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => {{1,2,6},{3},{4},{5},{7},{8}}
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => {{1,2,8},{3},{4},{5},{6},{7},{9}}
=> ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => {{1,2,10},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> ? = 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,12,11] => ?
=> ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => {{1,10,11},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,9,2,3,4,5,6,7,10,8] => ?
=> ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,10,9] => ?
=> ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,2,8,3,4,5,6,9,7] => ?
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,9,6] => ?
=> ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 2
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,2,3,7,4,5,8,6] => {{1},{2},{3},{4,7,8},{5},{6}}
=> ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [6,1,7,2,3,4,8,5] => ?
=> ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,8,7] => ?
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,8,6] => ?
=> ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,8,7] => {{1,4},{2},{3},{5},{6},{7,8}}
=> ? = 2
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,1,3,4,8,5,6,7] => {{1,2},{3},{4},{5,8},{6},{7}}
=> ? = 2
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,7,1,8,3,4,5,6] => {{1,2,7},{3},{4,8},{5},{6}}
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,1,3,9,4,5,6,7,8] => ?
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6,8] => ?
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,6,8,1,3,4,5,7] => ?
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,8,9,1,3,4,5,6,7] => ?
=> ? = 4
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2,1,10,3,4,5,6,7,8,9] => ?
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ?
=> ? = 3
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8,9] => ?
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8,10] => ?
=> ? = 3
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: St000069
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 90%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,6),(1,3),(1,6),(2,4),(2,6),(3,4),(3,5),(6,5)],7)
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4)],7)
=> 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
=> 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(5,4),(6,3),(6,5),(7,3),(7,4)],8)
=> ? = 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,6),(1,3),(1,6),(2,5),(2,6),(3,5),(5,4),(6,4)],7)
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(5,2),(5,3),(6,3),(6,4)],7)
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,5),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,1),(3,6)],7)
=> ? = 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(8,2)],9)
=> ? = 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,7),(0,8),(1,7),(1,8),(2,6),(2,8),(5,3),(6,3),(6,4),(7,5),(7,6),(8,4),(8,5)],9)
=> ? = 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(0,7),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4)],8)
=> ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(5,4),(6,2),(6,5),(7,3),(7,5)],8)
=> ? = 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,5),(1,7),(2,6),(2,7),(5,6),(6,3),(6,4),(7,3),(7,4)],8)
=> ? = 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(7,4),(7,5)],8)
=> ? = 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(3,5),(4,3),(5,7),(6,4),(7,9),(8,6),(9,1),(9,2)],10)
=> ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,10),(1,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7),(10,9)],11)
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,8),(0,9),(1,8),(1,9),(2,7),(2,9),(5,4),(6,3),(6,4),(7,3),(7,5),(8,6),(8,7),(9,5),(9,6)],10)
=> ? = 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,6),(0,8),(1,7),(1,8),(2,7),(2,8),(4,3),(5,3),(6,4),(7,5),(7,6),(8,4),(8,5)],9)
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(0,8),(1,6),(1,8),(2,4),(2,5),(5,3),(6,2),(6,7),(7,3),(7,4),(8,5),(8,7)],9)
=> ? = 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(0,7),(1,4),(1,7),(2,4),(2,7),(4,5),(5,6),(6,3),(7,6)],8)
=> ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6)],8)
=> ? = 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,7),(1,7),(2,5),(3,5),(3,6),(4,2),(4,6),(7,3),(7,4)],8)
=> ? = 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,4),(1,6),(4,5),(5,7),(6,7),(7,2),(7,3)],8)
=> ? = 2
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
=> ? = 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(3,6),(3,7)],8)
=> ? = 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,8),(1,5),(1,8),(2,5),(2,7),(5,6),(6,3),(6,4),(7,3),(7,4),(8,6),(8,7)],9)
=> ? = 2
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(2,7),(3,1),(3,7),(4,2),(4,3)],8)
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,1),(0,3),(1,2),(1,8),(2,6),(2,7),(3,7),(3,8),(7,4),(7,5),(8,4),(8,5),(8,6)],9)
=> ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,8),(1,7),(1,8),(2,7),(2,9),(6,4),(6,5),(7,3),(7,6),(8,6),(8,9),(9,3),(9,4),(9,5)],10)
=> ? = 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,10),(9,7),(10,1),(10,2)],11)
=> ? = 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ?
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ?
=> ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ?
=> ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ?
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ?
=> ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ?
=> ? = 2
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(5,6),(6,3),(7,6)],8)
=> ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(5,6)],8)
=> ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(4,3),(5,3),(6,2),(6,4),(7,4),(7,5)],8)
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,7),(4,5),(4,7)],8)
=> ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,7),(1,7),(3,6),(4,2),(4,6),(5,3),(5,4),(7,5)],8)
=> ? = 2
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(5,3),(6,2),(6,3)],7)
=> ? = 3
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,7),(1,4),(1,7),(4,6),(5,2),(5,3),(6,5),(7,6)],8)
=> ? = 2
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7)],8)
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,7),(2,1),(2,6),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ?
=> ? = 4
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,6),(2,7),(3,1),(3,6),(3,7),(4,5),(5,2),(5,3)],8)
=> ? = 3
Description
The number of maximal elements of a poset.
Matching statistic: St000068
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 80%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3),(6,4)],7)
=> 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,6),(1,3),(1,6),(2,4),(2,6),(3,4),(3,5),(6,5)],7)
=> 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,7),(1,6),(1,7),(2,5),(2,6),(5,4),(6,3),(6,4),(7,3),(7,5)],8)
=> ? = 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(8,2)],9)
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,7),(1,7),(4,6),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(5,4),(6,3),(6,5),(7,3),(7,4)],8)
=> ? = 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,8),(1,7),(1,8),(2,6),(2,7),(5,4),(6,3),(6,4),(7,3),(7,5),(8,5),(8,6)],9)
=> ? = 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(3,5),(4,3),(5,7),(6,4),(7,9),(8,6),(9,1),(9,2)],10)
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,8),(1,8),(4,5),(5,7),(6,4),(7,2),(7,3),(8,6)],9)
=> ? = 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
=> ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,7),(1,4),(4,7),(5,6),(6,2),(6,3),(7,5)],8)
=> ? = 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
=> ? = 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,6),(1,7),(2,5),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4)],8)
=> ? = 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,6),(3,5),(3,7),(6,4),(6,5),(7,4)],8)
=> ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,7),(0,8),(1,7),(1,8),(2,6),(2,8),(5,3),(6,3),(6,4),(7,5),(7,6),(8,4),(8,5)],9)
=> ? = 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,9),(1,8),(1,9),(2,7),(2,8),(5,4),(6,3),(6,4),(7,3),(7,5),(8,5),(8,6),(9,6),(9,7)],10)
=> ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,10),(9,7),(10,1),(10,2)],11)
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,9),(1,9),(4,6),(5,4),(6,8),(7,5),(8,2),(8,3),(9,7)],10)
=> ? = 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2),(8,6)],9)
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,8),(1,4),(4,8),(5,7),(6,5),(7,2),(7,3),(8,6)],9)
=> ? = 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
=> ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,7),(1,7),(2,7),(5,6),(6,3),(6,4),(7,5)],8)
=> ? = 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,7),(1,4),(1,7),(4,6),(5,2),(5,3),(6,5),(7,6)],8)
=> ? = 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> ? = 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,6),(1,5),(2,5),(5,6),(6,3),(6,4)],7)
=> ? = 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,5),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,7),(4,3),(5,4),(5,7),(6,3),(6,7)],8)
=> ? = 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(6,4),(6,5),(7,5)],8)
=> ? = 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,7),(1,8),(2,7),(2,8),(3,6),(3,8),(6,5),(7,4),(7,6),(8,4),(8,5)],9)
=> ? = 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,6),(2,4),(2,5),(3,1),(3,5),(5,6)],7)
=> ? = 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(0,7),(1,4),(1,6),(2,3),(2,4),(3,6),(3,7),(4,5),(4,7),(6,5)],8)
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,7),(0,8),(1,7),(1,8),(2,3),(2,7),(3,6),(3,8),(6,4),(7,5),(7,6),(8,4),(8,5)],9)
=> ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,8),(0,9),(1,8),(1,9),(2,7),(2,9),(5,4),(6,3),(6,4),(7,3),(7,5),(8,6),(8,7),(9,5),(9,6)],10)
=> ? = 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,10),(1,9),(1,10),(2,8),(2,9),(5,4),(6,3),(6,5),(7,3),(7,4),(8,5),(8,7),(9,6),(9,7),(10,6),(10,8)],11)
=> ? = 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ?
=> ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,9),(1,4),(4,9),(5,6),(6,8),(7,5),(8,2),(8,3),(9,7)],10)
=> ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ?
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ?
=> ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 2
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
=> ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,4),(6,5),(7,5)],8)
=> ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(0,5),(3,7),(4,7),(5,3),(6,1),(6,2),(7,6)],8)
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,7),(1,6),(2,6),(3,4),(3,5),(6,7),(7,3)],8)
=> ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,4),(1,6),(4,5),(5,7),(6,7),(7,2),(7,3)],8)
=> ? = 2
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,6),(1,2),(1,3),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 2
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,6),(1,6),(2,3),(3,6),(6,4),(6,5)],7)
=> ? = 3
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ? = 3
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7)
=> ? = 2
Description
The number of minimal elements in a poset.
Matching statistic: St000654
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 2
[2]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [4,3,2,1,5] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [5,4,3,2,1,6] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> [1,6,5,4,3,2] => 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [6,5,4,3,2,1,7] => ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [1,5,4,3,2,6] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,2,1,5,4] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
=> [1,2,6,5,4,3] => 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ? = 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [7,6,5,4,3,2,1,8] => ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [2,1,5,4,3,6] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [1,5,4,3,2,6] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [1,3,2,6,5,4] => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [1,2,6,5,4,3] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> [1,2,7,6,5,4,3] => 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0]]
=> [1,8,7,6,5,4,3,2] => ? = 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [8,7,6,5,4,3,2,1,9] => ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,1,-1,1,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,6,5,4,3,2,8] => ? = 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [3,2,1,5,4,6] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [1,2,5,4,3,6] => 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [1,5,4,3,2,6] => 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [4,3,2,1,6,5] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
=> [1,2,8,7,6,5,4,3] => ? = 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0]]
=> [1,9,8,7,6,5,4,3,2] => ? = 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> [9,8,7,6,5,4,3,2,1,10] => ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,7,6,5,4,3,2,9] => ? = 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [2,1,7,6,5,4,3,8] => ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,1,-1,0,1,0,0,0],[1,-1,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,6,5,4,3,2,8] => ? = 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,1,0,-1,1,0],[0,0,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> [1,3,2,8,7,6,5,4] => ? = 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,1,-1,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
=> [1,2,8,7,6,5,4,3] => ? = 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> [1,2,9,8,7,6,5,4,3] => ? = 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0]]
=> [1,10,9,8,7,6,5,4,3,2] => ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1]]
=> [10,9,8,7,6,5,4,3,2,1,11] => ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> [1,9,8,7,6,5,4,3,2,10] => ? = 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0,0,0],[0,1,0,-1,1,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [2,1,8,7,6,5,4,3,9] => ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,-1,0,1,0],[0,0,0,1,-1,0,1,0,0],[0,0,1,-1,0,1,0,0,0],[0,1,-1,0,1,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,7,6,5,4,3,2,9] => ? = 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,-1,1,0],[0,1,0,0,-1,1,0,0],[1,0,0,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [3,2,1,7,6,5,4,8] => ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,-1,1,0],[0,1,-1,1,-1,1,0,0],[1,-1,1,-1,1,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,7,6,5,4,3,8] => ? = 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,1,-1,0,0,1,0,0],[1,-1,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,6,5,4,3,2,8] => ? = 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [4,3,2,1,6,5,7] => ? = 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [5,4,3,2,1,7,6] => ? = 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,1,0,0,-1,1,0],[0,1,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0]]
=> [1,4,3,2,8,7,6,5] => ? = 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,1,-1,1,-1,1],[0,0,1,-1,1,-1,1,0],[0,1,-1,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> [1,2,3,8,7,6,5,4] => ? = 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,-1,1],[0,0,0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0,0,0],[0,1,0,-1,1,0,0,0,0],[0,0,0,1,0,0,0,0,0]]
=> [1,3,2,9,8,7,6,5,4] => ? = 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> [2,1,7,6,5,4,3] => ? = 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,0,1],[0,0,1,-1,0,0,1,0],[0,1,-1,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
=> [1,2,8,7,6,5,4,3] => ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,-1,0,1],[0,0,0,0,1,-1,0,1,0],[0,0,0,1,-1,0,1,0,0],[0,0,1,-1,0,1,0,0,0],[0,1,-1,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> [1,2,9,8,7,6,5,4,3] => ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0]]
=> [1,2,10,9,8,7,6,5,4,3] => ? = 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0]]
=> [1,11,10,9,8,7,6,5,4,3,2] => ? = 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1]]
=> ? => ? = 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1]]
=> ? => ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? => ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,0,1,0],[0,0,0,0,1,-1,0,1,0,0],[0,0,0,1,-1,0,1,0,0,0],[0,0,1,-1,0,1,0,0,0,0],[0,1,-1,0,1,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? => ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,-1,1,0],[0,0,1,0,0,-1,1,0,0],[0,1,0,0,-1,1,0,0,0],[1,0,0,-1,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? => ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,1,-1,1,-1,1,0],[0,0,1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? => ? = 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0,0,0],[0,0,0,1,-1,0,0,1,0],[0,0,1,-1,0,0,1,0,0],[0,1,-1,0,0,1,0,0,0],[1,-1,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? => ? = 2
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,-1,1,0],[1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1]]
=> [4,3,2,1,7,6,5,8] => ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,1,-1,1,0,-1,1,0],[1,-1,1,0,-1,1,0,0],[0,1,0,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? = 2
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000740The last entry of a permutation. St000989The number of final rises of a 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. St000990The first ascent of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000025The number of initial rises of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000193The row of the unique '1' in the first column of the alternating sign matrix.
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