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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000994
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000352
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[] => []
=> [] => ? = 0
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000814
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000814: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000814: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [2,1] => [1,1]
=> 2 = 1 + 1
[2,1] => [1,1,0,0]
=> [1,2] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 3 = 2 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 2 = 1 + 1
[2,4,5,3,7,8,6,1] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3 + 1
[2,4,3,5,6,8,7,1] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3 + 1
[2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3 + 1
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3 + 1
[1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3 + 1
[4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3 + 1
[3,4,2,5,6,1,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3 + 1
[4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4 + 1
[1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ?
=> ? = 3 + 1
[2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7,8] => ?
=> ? = 3 + 1
[4,2,3,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4 + 1
[4,2,3,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3 + 1
[2,4,5,1,7,8,3,6] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3 + 1
[2,4,1,5,6,8,3,7] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3 + 1
[2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3 + 1
[3,4,1,5,6,2,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3 + 1
[1,3,4,5,7,8,2,6] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3 + 1
[4,1,2,5,6,3,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3 + 1
[4,1,2,3,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4 + 1
[1,2,3,5,7,8,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,7,8,5,6,9,10] => ?
=> ? = 4 + 1
[4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,1,2,7,8,3,4,9,10] => ?
=> ? = 4 + 1
[4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,7,8,9,10,3,4] => ?
=> ? = 4 + 1
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[3,1,5,2,7,4,9,6,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4 + 1
[2,3,4,6,1,7,8,9,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3 + 1
[6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,1,3,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,1,5,4,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2 + 1
[3,4,5,6,7,8,9,10,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4 + 1
[2,4,5,1,7,8,6,3] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3 + 1
[7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2 + 1
[2,1,4,3,6,5,8,7,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,8,9,6] => ?
=> ? = 4 + 1
[6,5,4,9,8,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,8,9,4,5,6] => ?
=> ? = 3 + 1
[5,4,7,6,9,8,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,2,3,6,7,4,5,8,9] => ?
=> ? = 2 + 1
[3,2,1,6,5,4,9,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3,7,8,9] => ?
=> ? = 3 + 1
[3,2,5,4,7,6,9,8,1] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4 + 1
[7,6,5,10,9,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,8,9,10,5,6,7] => ?
=> ? = 3 + 1
[7,8,9,10,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2 + 1
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
Matching statistic: St001176
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1]
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => [1,1]
=> 1
[2,1] => [1,1,0,0]
=> [1,2] => [2]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1
[2,4,5,3,7,8,6,1] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,4,3,5,6,8,7,1] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3
[2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[3,4,2,5,6,1,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3
[4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ?
=> ? = 3
[2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7,8] => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[4,2,3,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[2,4,5,1,7,8,3,6] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,4,1,5,6,8,3,7] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3
[2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[3,4,1,5,6,2,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3
[1,3,4,5,7,8,2,6] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[4,1,2,5,6,3,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[4,1,2,3,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,2,3,5,7,8,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,7,8,5,6,9,10] => ?
=> ? = 4
[4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,1,2,7,8,3,4,9,10] => ?
=> ? = 4
[4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,7,8,9,10,3,4] => ?
=> ? = 4
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[3,1,5,2,7,4,9,6,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4
[2,3,4,6,1,7,8,9,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[] => []
=> [] => []
=> ? = 0
[6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,3,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,5,4,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[3,4,5,6,7,8,9,10,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[2,4,5,1,7,8,6,3] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
[2,1,4,3,6,5,8,7,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,8,9,6] => ?
=> ? = 4
[6,5,4,9,8,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,8,9,4,5,6] => ?
=> ? = 3
[5,4,7,6,9,8,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,2,3,6,7,4,5,8,9] => ?
=> ? = 2
[3,2,1,6,5,4,9,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3,7,8,9] => ?
=> ? = 3
[3,2,5,4,7,6,9,8,1] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4
[7,6,5,10,9,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,8,9,10,5,6,7] => ?
=> ? = 3
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000185
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1]
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => [1,1]
=> 1
[2,1] => [1,1,0,0]
=> [1,2] => [2]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [3]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1
[2,4,5,3,7,8,6,1] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,4,3,5,6,8,7,1] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3
[2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[3,4,2,5,6,1,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3
[4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ?
=> ? = 3
[2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7,8] => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[4,2,3,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[2,4,5,1,7,8,3,6] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,4,1,5,6,8,3,7] => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3
[2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[3,4,1,5,6,2,8,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 3
[1,3,4,5,7,8,2,6] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[4,1,2,5,6,3,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[4,1,2,3,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,2,3,5,7,8,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,7,8,5,6,9,10] => ?
=> ? = 4
[4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,1,2,7,8,3,4,9,10] => ?
=> ? = 4
[4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,7,8,9,10,3,4] => ?
=> ? = 4
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[3,1,5,2,7,4,9,6,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4
[2,3,4,6,1,7,8,9,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,3,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,5,4,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[3,4,5,6,7,8,9,10,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[2,4,5,1,7,8,6,3] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[10,1,2,3,4,5,6,7,8,11,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [10,1]
=> ? = 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [10,1]
=> ? = 1
[7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [10,1]
=> ? = 1
[1,11,2,3,4,5,6,7,8,9,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [10,1]
=> ? = 1
[2,1,4,3,6,5,8,7,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,8,9,6] => ?
=> ? = 4
[10,9,8,7,6,5,4,3,1,11,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> ? = 1
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St000336
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[2,1] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[2,4,5,3,7,8,6,1] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[4,2,3,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[2,4,5,1,7,8,3,6] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,3,4,5,7,8,2,6] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[4,1,2,5,6,3,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ?
=> ? = 3
[4,1,2,3,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ?
=> ? = 4
[1,2,3,5,7,8,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ?
=> ? = 3
[2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,7,8,5,6,9,10] => ?
=> ? = 4
[4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,1,2,7,8,3,4,9,10] => ?
=> ? = 4
[4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,7,8,9,10,3,4] => ?
=> ? = 4
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[7,1,2,3,4,5,9,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 2
[3,1,5,2,7,4,9,6,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,8,9,6,7] => ?
=> ? = 4
[7,6,1,2,3,4,9,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 2
[7,8,9,1,6,2,3,4,5] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[2,3,4,6,1,7,8,9,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[7,8,9,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[7,3,4,5,6,1,9,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 2
[2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6,9,8] => [[1,2,3,6,8],[4,5,7,9]]
=> ? = 4
[6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,3,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,1,5,4,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,9,8,10] => ?
=> ? = 2
[2,3,4,5,6,8,9,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,7,4,6,9,8] => [[1,2,3,5,8],[4,6,7,9]]
=> ? = 4
[3,4,5,6,7,8,9,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,9] => [[1,2,3,5,7,9],[4,6,8]]
=> ? = 3
[2,3,4,5,6,7,8,10,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6,9,8,10] => [[1,2,3,6,8,10],[4,5,7,9]]
=> ? = 4
[2,4,5,1,7,8,6,3] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[8,7,6,5,4,3,9,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[7,8,9,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[8,7,6,5,3,4,9,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[10,1,2,3,4,5,6,7,8,11,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [[1,2,4,5,6,7,8,9,10,11],[3]]
=> ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
Description
The leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Matching statistic: St000183
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> 0
[1,2] => [1,0,1,0]
=> [1]
=> 1
[2,1] => [1,1,0,0]
=> []
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> []
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[2,1,4,3,6,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> ? = 3
[3,1,5,2,4,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 2
[3,1,5,4,2,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 2
[3,2,5,1,4,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 2
[3,2,5,4,1,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 2
[3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ? = 3
[3,2,4,5,6,7,1,8] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2]
=> ? = 3
[2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 3
[5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,3,4,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,4,2,3,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,3,2,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[3,4,2,5,1,6,7,8] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 3
[2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 3
[5,4,3,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,3,4,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,4,2,1,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,4,1,2,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,3,2,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,2,3,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,2,1,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 3
[4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[4,2,3,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[4,3,1,2,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[4,2,1,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[4,1,2,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 4
[2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 4
[3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 4
[2,4,5,3,7,8,6,1] => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,1,1]
=> ? = 3
[2,3,5,6,7,4,8,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,2,1]
=> ? = 3
[2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,2,1]
=> ? = 3
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1,1]
=> ? = 3
[1,3,5,6,7,8,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 3
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 3
[3,2,1,6,5,8,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3]
=> ? = 3
[1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 3
[3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,1]
=> ? = 4
[2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1,1]
=> ? = 4
[3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2]
=> ? = 4
[2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,1]
=> ? = 4
[2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2,2]
=> ? = 4
[1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,2,1]
=> ? = 3
[1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,1,1,1]
=> ? = 4
[3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 4
[2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,1]
=> ? = 4
[4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3]
=> ? = 3
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000035
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,2] => [1,0,1,0]
=> [2,1] => [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 1
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 3
[1,3,7,2,4,5,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,2,4,6,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,2,5,4,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,2,5,6,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,2,6,4,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,2,6,5,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,2,5,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,2,6,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,5,2,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,5,6,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,6,2,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,4,6,5,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,2,4,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,2,6,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,4,2,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,4,6,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,6,2,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,5,6,4,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,2,4,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,2,5,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,4,2,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,4,5,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,5,2,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,3,7,6,5,4,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,1,3,5,2,4,6] => ? = 2
[1,4,6,7,2,3,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,6,7,2,5,3] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,6,7,3,2,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,6,7,3,5,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,6,7,5,2,3] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,6,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,1,4,2,3,5,6] => ? = 2
[1,4,7,2,3,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,2,3,6,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,2,5,3,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,2,5,6,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,2,6,3,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,2,6,5,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,2,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,2,6,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,5,2,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,5,6,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,6,2,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,3,6,5,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,2,3,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,2,6,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,3,2,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,3,6,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,6,2,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,5,6,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
[1,4,7,6,2,3,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,2,7,1,3,5,6] => ? = 2
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St001597
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%
Values
[1] => [1,0]
=> []
=> [[],[]]
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [[1],[]]
=> 1
[2,1] => [1,1,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[3,2,1] => [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 0
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? = 3
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[4,4,3,2,1],[]]
=> ? = 3
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? = 3
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ? = 3
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? = 3
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000251
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1] => {{1}}
=> ? = 0
[1,2] => [1,0,1,0]
=> [2,1] => {{1,2}}
=> 1
[2,1] => [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => {{1,3},{2}}
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => {{1,4},{2,3}}
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 2
[8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 1
[7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 1
[7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 1
[7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 1
[7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 1
[4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,6,7,8] => {{1,5},{2,4},{3},{6},{7},{8}}
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 2
[8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,4,5,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,5,3,4,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,7,4,3,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,3,4,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
Description
The number of nonsingleton blocks of a set partition.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000829The Ulam distance of a permutation to the identity permutation. St001874Lusztig's a-function for the symmetric group. St000731The number of double exceedences of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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