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Your data matches 156 different statistics following compositions of up to 3 maps.
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Matching statistic: St000142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[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]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
Description
The number of even parts of a partition.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000254
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => {{1,3},{2}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => {{1,4},{2,3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => {{1,4},{2,3,5}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => {{1,3},{2,4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000288
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[1,0,1,1,0,0]
=> [2,3,1] => 10 => 1
[1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[1,1,0,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 100 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 110 => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 100 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 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]
=> [2,1,4,3,5] => 1010 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1010 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 1000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 1000 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1100 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 1000 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1100 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 1100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 1000 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0100 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1100 => 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000389
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,1,3] => 10 => 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]
=> [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 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]
=> [2,1,4,3,5] => 1010 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[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]
=> [3,1,4,2,5] => 1010 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 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]
=> [3,1,5,2,4] => 1010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,1,3] => 10 => 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]
=> [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 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]
=> [2,1,4,3,5] => 1010 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[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]
=> [3,1,4,2,5] => 1010 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 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]
=> [3,1,5,2,4] => 1010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000566
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[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]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St001251
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[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]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[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]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[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]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
The following 146 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. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000377The dinv defect of an integer partition. St000877The depth of the binary word interpreted as a path. St000919The number of maximal left branches of a binary tree. St001176The size of a partition minus its first part. St000010The length of the partition. St000097The order of the largest clique of the graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000507The number of ascents of a standard tableau. St000814The 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. St001581The achromatic number of a graph. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000011The number of touch points (or returns) of a Dyck path. St000098The chromatic number of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000806The semiperimeter of the associated bargraph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000292The number of ascents of a binary word. St001214The aft of an integer partition. St000568The hook number of a binary tree. St000306The bounce count of a Dyck path. St000167The number of leaves of an ordered tree. St000884The number of isolated descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys 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. St000354The number of recoils of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000035The number of left outer peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000386The number of factors DDU in a Dyck path. St000834The number of right outer peaks of a permutation. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000632The jump number of the poset. St000021The number of descents of a permutation. St001665The number of pure excedances of a permutation. St000325The width of the tree associated to a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St001427The number of descents of a signed permutation. St000702The number of weak deficiencies of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. 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. 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. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001812The biclique partition number of a graph. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a 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. St000822The Hadwiger number of the graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000619The number of cyclic descents of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000054The first entry of the permutation. St000353The number of inner valleys of a permutation. St001330The hat guessing number of a graph. St001728The number of invisible descents of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000092The number of outer peaks of a permutation. 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 $eA$. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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