Your data matches 112 different statistics following compositions of up to 3 maps.
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Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000035: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
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''.
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00109: Permutations descent wordBinary words
St000390: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => => ? = 0
[1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => 11 => 1
[1,0,1,1,0,0]
=> [2,3,1] => 01 => 1
[1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[1,1,0,1,0,0]
=> [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 111 => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 011 => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 110 => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 110 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 110 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1011 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 1011 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 1101 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1110 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0110 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 1110 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0110 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1110 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1110 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 0110 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1100 => 1
[]
=> [] => ? => ? = 0
Description
The number of runs of ones in a binary word.
Mp00102: Dyck paths rise compositionInteger compositions
Mp00039: Integer compositions complementInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,0,1,0]
=> [1,1] => [2] => [2]
=> 1
[1,1,0,0]
=> [2] => [1,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,3] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [4,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [3,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [3,2] => [3,2]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [3,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,3] => [3,2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,3] => [3,2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,4] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,4] => [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => [2,1,1,1]
=> 1
[]
=> [] => [] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ? => ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ? => ?
=> ? = 4
Description
The number of parts of an integer partition that are at least two.
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00131: Permutations descent bottomsBinary words
St000288: Binary words ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 10 => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 10 => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1100 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1001 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 0110 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1010 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1100 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ? => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ? => ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ? => ? = 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ? => ? = 3
[]
=> []
=> [] => => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ? => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ? => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ? => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ? => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ? => ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ? => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ? => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ? => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ? => ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ? => ? = 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00109: Permutations descent wordBinary words
St000389: Binary words ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 01 => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 10 => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 001 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 001 => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0001 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1001 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0101 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 0101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0010 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0010 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ? => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ? => ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ? => ? = 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ? => ? = 2
[]
=> []
=> [] => ? => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ? => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ? => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ? => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ? => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ? => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ? => ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ? => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ? => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ? => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ? => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ? => ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ? => ? = 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000142
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00204: Permutations LLPSInteger partitions
St000142: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ?
=> ? = 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ?
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ?
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ?
=> ? = 2
Description
The number of even parts of a partition.
Matching statistic: St001251
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00204: Permutations LLPSInteger partitions
St001251: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ?
=> ? = 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ?
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ?
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ?
=> ? = 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00204: Permutations LLPSInteger partitions
St001252: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ?
=> ? = 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ?
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ?
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ?
=> ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ?
=> ? = 2
Description
Half the sum of the even parts of a partition.
Matching statistic: St000345
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00204: Permutations LLPSInteger partitions
St000345: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ?
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ?
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ?
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ?
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ?
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ? = 4 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ?
=> ? = 2 + 1
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
Mp00028: Dyck paths reverseDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00204: Permutations LLPSInteger partitions
St000935: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,5,4,8,6,7] => ?
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,7,8,9,2,6] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,7,8,9,2,5] => ?
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,8,5,7,9] => ?
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,3,6,4,5,7,8,9] => ?
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,8,3,4,5,7,9] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,2,7,3,4,5,6,8,9] => ?
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,8,9] => ?
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,4,1,5,7,6,8,9] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,7,6,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9] => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ? = 4 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,8,7,9,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,7,6,8,9,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,7,6,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,6,5,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,5,4,6,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,7,4,5,6,8,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,4,2,3,5,6,7,8,9,10] => ?
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,9,4,6,8,10] => ?
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,8,6,7,9,10] => ?
=> ? = 2 + 1
Description
The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001657The number of twos in an integer partition. St001389The number of partitions of the same length below the given integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000291The number of descents of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000157The number of descents of a standard tableau. St000292The number of ascents of a binary word. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St000340The number of non-final maximal constant sub-paths of length greater than one. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000374The number of exclusive right-to-left minima of a permutation. St000251The number of nonsingleton blocks of a set partition. St000670The reversal length of a permutation. St000662The staircase size of the code of a permutation. St000386The number of factors DDU in a Dyck path. St000354The number of recoils of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001512The minimum rank of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001354The number of series nodes in the modular decomposition of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000672The number of minimal elements in Bruhat order not less than the permutation. St000167The number of leaves of an ordered tree. St001712The number of natural descents of a standard Young tableau. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000201The number of leaf nodes in a binary tree. St000568The hook number of a binary tree. St001427The number of descents of a signed permutation. St000021The number of descents of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000647The number of big descents of a permutation. St001469The holeyness of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St000099The number of valleys of a permutation, including the boundary. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St001896The number of right descents of a signed permutations. St001722The number of minimal chains with small intervals between a binary word and the top element. St001597The Frobenius rank of a skew partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000807The sum of the heights of the valleys of the associated bargraph. St001470The cyclic holeyness of a permutation. St000805The number of peaks of the associated bargraph. St001487The number of inner corners of a skew partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001624The breadth of a lattice.