Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001568
Mapping
St001568: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 1
[1,1]
 => 2
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 2
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 2
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 1
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 2
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 1
[3,2,1]
 => 1
[3,1,1,1]
 => 2
[2,2,2]
 => 1
[2,2,1,1]
 => 3
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 2
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 2
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 2
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 3
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 2
[8]
 => 1
[7,1]
 => 1
[6,2]
 => 1
[6,1,1]
 => 2
[5,3]
 => 1
[5,2,1]
 => 1
[5,1,1,1]
 => 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000352
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,1,4,2] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,4,2,1,3] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,6,1,4,5,2] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,1,2,4] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [4,1,5,2,3] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,5,2,1,4,3] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => [3,7,1,4,5,6,2] => ? = 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [8,2,3,4,5,6,1,7] => ? = 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => [5,1,7,3,4,2,6] => ? = 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,4,6,2,1,5] => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,5,1,2,6,4] => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,6,1,5,3] => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [5,4,6,1,2,3] => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,6,2,1,4,5,3] => ? = 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => [3,8,1,4,5,6,7,2] => ? = 2
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => [9,2,3,4,5,6,7,1,8] => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => [6,1,8,3,4,5,2,7] => ? = 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => [5,2,1,7,4,3,6] => ? = 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [4,7,5,3,1,2,6] => ? = 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,2,6,3,5] => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,6,1,4,2,5] => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,5,6,2,4] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [6,2,5,3,1,4] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [6,5,2,3,1,4] => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,6,1,2,7,5,4] => ? = 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,5,7,1,4,2,3] => ? = 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [8,1,4,5,6,7,2,3] => [8,7,2,1,4,5,6,3] => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,9,2] => [3,9,1,4,5,6,7,8,2] => ? = 2
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,9] => [10,2,3,4,5,6,7,8,1,9] => ? = 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,4,5,6,7,1,2,8] => [7,1,9,3,4,5,6,2,8] => ? = 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [2,8,4,5,6,1,3,7] => [6,2,1,8,4,5,3,7] => ? = 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,3,4,5,1,8,2,7] => [5,8,6,3,4,1,2,7] => ? = 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => [5,2,3,1,7,4,6] => ? = 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [4,1,2,5,7,3,6] => ? = 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [3,4,7,2,5,1,6] => ? = 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,6,1,3,5] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,4,2,1,3,5] => 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,6,1,4,2,7,5] => ? = 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [5,2,1,6,3,4] => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,5,1,6,2,4] => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,1,5,3,2,4] => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,5,2,3,6,1,4] => ? = 2
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,8,5,6,7,2,4] => [3,7,1,2,8,5,6,4] => ? = 2
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [4,2,7,1,5,6,3] => ? = 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,4,7,1,2,6,3] => ? = 3
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,6,8,1,4,5,2,3] => ? = 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [9,1,4,5,6,7,8,2,3] => [9,8,2,1,4,5,6,7,3] => ? = 2
[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]
 => [3,1,4,5,6,7,8,9,10,2] => [3,10,1,4,5,6,7,8,9,2] => ? = 2
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [11,2,3,4,5,6,7,8,9,1,10] => ? = 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,4,5,6,7,8,1,2,9] => [8,1,10,3,4,5,6,7,2,9] => ? = 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [2,9,4,5,6,7,1,3,8] => [7,2,1,9,4,5,6,3,8] => ? = 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [7,3,4,5,6,1,9,2,8] => [6,9,7,3,4,5,1,2,8] => ? = 2
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [2,3,8,5,6,1,4,7] => [6,2,3,1,8,5,4,7] => ? = 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [8,6,4,5,1,2,3,7] => [5,1,2,6,4,8,3,7] => ? = 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [5,3,4,1,6,8,2,7] => [4,8,5,3,1,6,2,7] => ? = 2
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [5,1,7,2,3,4,6] => ? = 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [2,5,4,1,7,3,6] => [4,2,5,7,3,1,6] => ? = 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,7,1,5,2,3,6] => ? = 2
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,7,1,4,5,2,6] => ? = 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => [4,1,6,3,7,2,5] => ? = 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,6,1,2,4,5] => 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [4,1,6,2,3,5] => 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,6,2,1,3,4,5] => ? = 2
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [3,1,4,8,6,7,2,5] => [3,7,1,4,2,8,6,5] => ? = 2
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => [5,2,3,7,1,6,4] => ? = 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [5,1,2,6,3,4] => 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,5,1,6,7,4,2] => ? = 2
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [7,2,6,3,1,5,4] => ? = 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [7,5,6,3,1,2,4] => ? = 3
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => [8,6,2,3,7,5,1,4] => ? = 2
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,9,5,6,7,8,2,4] => [3,8,1,2,9,5,6,7,4] => ? = 2
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,2,5,6,3] => ? = 1
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,1,4,5,2,7,8,3] => [6,5,8,1,4,2,7,3] => ? = 3
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
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.