Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001280
(load all 7 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 0
[1,2] => [1,1]
 => 0
[2,1] => [2]
 => 1
[1,2,3] => [1,1,1]
 => 0
[1,3,2] => [2,1]
 => 1
[2,1,3] => [2,1]
 => 1
[2,3,1] => [3]
 => 1
[3,1,2] => [3]
 => 1
[3,2,1] => [2,1]
 => 1
[1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => 1
[1,3,2,4] => [2,1,1]
 => 1
[1,3,4,2] => [3,1]
 => 1
[1,4,2,3] => [3,1]
 => 1
[1,4,3,2] => [2,1,1]
 => 1
[2,1,3,4] => [2,1,1]
 => 1
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => 1
[2,3,4,1] => [4]
 => 1
[2,4,1,3] => [4]
 => 1
[2,4,3,1] => [3,1]
 => 1
[3,1,2,4] => [3,1]
 => 1
[3,1,4,2] => [4]
 => 1
[3,2,1,4] => [2,1,1]
 => 1
[3,2,4,1] => [3,1]
 => 1
[3,4,1,2] => [2,2]
 => 2
[3,4,2,1] => [4]
 => 1
[4,1,2,3] => [4]
 => 1
[4,1,3,2] => [3,1]
 => 1
[4,2,1,3] => [3,1]
 => 1
[4,2,3,1] => [2,1,1]
 => 1
[4,3,1,2] => [4]
 => 1
[4,3,2,1] => [2,2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => 2
[1,3,4,2,5] => [3,1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => 1
[1,3,5,2,4] => [4,1]
 => 1
[1,3,5,4,2] => [3,1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001737
(load all 13 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [3,2,1] => 1
[3,2,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => [4,3,1,2] => 1
[2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => [3,2,1,4] => 1
[3,1,4,2] => [4,2,1,3] => 1
[3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => [4,1,3,2] => 1
[4,1,2,3] => [4,3,2,1] => 1
[4,1,3,2] => [3,4,2,1] => 1
[4,2,1,3] => [2,4,3,1] => 1
[4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [4,2,3,1] => 1
[4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => [1,5,4,2,3] => 1
[1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,3,2,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => [1,4,2,5,3] => 2
Description
The number of descents of type 2 in a permutation. A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000010
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => 1 = 0 + 1
[1,2] => [1,1]
 => [2]
 => 1 = 0 + 1
[2,1] => [2]
 => [1,1]
 => 2 = 1 + 1
[1,2,3] => [1,1,1]
 => [3]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[2,3,1] => [3]
 => [2,1]
 => 2 = 1 + 1
[3,1,2] => [3]
 => [2,1]
 => 2 = 1 + 1
[3,2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[1,3,2,4] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[1,3,4,2] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[1,4,2,3] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[1,4,3,2] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[2,1,4,3] => [2,2]
 => [2,1,1]
 => 3 = 2 + 1
[2,3,1,4] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[2,3,4,1] => [4]
 => [3,1]
 => 2 = 1 + 1
[2,4,1,3] => [4]
 => [3,1]
 => 2 = 1 + 1
[2,4,3,1] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[3,1,2,4] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[3,1,4,2] => [4]
 => [3,1]
 => 2 = 1 + 1
[3,2,1,4] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[3,2,4,1] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[3,4,1,2] => [2,2]
 => [2,1,1]
 => 3 = 2 + 1
[3,4,2,1] => [4]
 => [3,1]
 => 2 = 1 + 1
[4,1,2,3] => [4]
 => [3,1]
 => 2 = 1 + 1
[4,1,3,2] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[4,2,1,3] => [3,1]
 => [2,2]
 => 2 = 1 + 1
[4,2,3,1] => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
[4,3,1,2] => [4]
 => [3,1]
 => 2 = 1 + 1
[4,3,2,1] => [2,2]
 => [2,1,1]
 => 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,3,4,5,2] => [4,1]
 => [3,2]
 => 2 = 1 + 1
[1,3,5,2,4] => [4,1]
 => [3,2]
 => 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,4,2,5,3] => [4,1]
 => [3,2]
 => 2 = 1 + 1
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
 => [3,2]
 => 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
 => [3,1,1]
 => 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => []
 => 0
[1,2] => [1,1]
 => [2]
 => []
 => 0
[2,1] => [2]
 => [1,1]
 => [1]
 => 1
[1,2,3] => [1,1,1]
 => [3]
 => []
 => 0
[1,3,2] => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3] => [2,1]
 => [2,1]
 => [1]
 => 1
[2,3,1] => [3]
 => [1,1,1]
 => [1,1]
 => 1
[3,1,2] => [3]
 => [1,1,1]
 => [1,1]
 => 1
[3,2,1] => [2,1]
 => [2,1]
 => [1]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [4]
 => []
 => 0
[1,2,4,3] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2,2]
 => [2]
 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,3,1] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,2,1,4] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2] => [2,2]
 => [2,2]
 => [2]
 => 2
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,1,3,2] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[4,2,1,3] => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[4,2,3,1] => [2,1,1]
 => [3,1]
 => [1]
 => 1
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,3,2,1] => [2,2]
 => [2,2]
 => [2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => []
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => [2]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [3,2]
 => [2]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000291
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => [1] => 1 => 0
[1,2] => {{1},{2}}
 => [1,1] => 11 => 0
[2,1] => {{1,2}}
 => [2] => 10 => 1
[1,2,3] => {{1},{2},{3}}
 => [1,1,1] => 111 => 0
[1,3,2] => {{1},{2,3}}
 => [1,2] => 110 => 1
[2,1,3] => {{1,2},{3}}
 => [2,1] => 101 => 1
[2,3,1] => {{1,2,3}}
 => [3] => 100 => 1
[3,1,2] => {{1,2,3}}
 => [3] => 100 => 1
[3,2,1] => {{1,3},{2}}
 => [2,1] => 101 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1111 => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,1,2] => 1110 => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,2,1] => 1101 => 1
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3] => 1100 => 1
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3] => 1100 => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,2,1] => 1101 => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,1] => 1011 => 1
[2,1,4,3] => {{1,2},{3,4}}
 => [2,2] => 1010 => 2
[2,3,1,4] => {{1,2,3},{4}}
 => [3,1] => 1001 => 1
[2,3,4,1] => {{1,2,3,4}}
 => [4] => 1000 => 1
[2,4,1,3] => {{1,2,3,4}}
 => [4] => 1000 => 1
[2,4,3,1] => {{1,2,4},{3}}
 => [3,1] => 1001 => 1
[3,1,2,4] => {{1,2,3},{4}}
 => [3,1] => 1001 => 1
[3,1,4,2] => {{1,2,3,4}}
 => [4] => 1000 => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => [2,1,1] => 1011 => 1
[3,2,4,1] => {{1,3,4},{2}}
 => [3,1] => 1001 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => [2,2] => 1010 => 2
[3,4,2,1] => {{1,2,3,4}}
 => [4] => 1000 => 1
[4,1,2,3] => {{1,2,3,4}}
 => [4] => 1000 => 1
[4,1,3,2] => {{1,2,4},{3}}
 => [3,1] => 1001 => 1
[4,2,1,3] => {{1,3,4},{2}}
 => [3,1] => 1001 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => [2,1,1] => 1011 => 1
[4,3,1,2] => {{1,2,3,4}}
 => [4] => 1000 => 1
[4,3,2,1] => {{1,4},{2,3}}
 => [2,2] => 1010 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,1,2,1] => 11101 => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,1,3] => 11100 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,1,3] => 11100 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => 11101 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => 11011 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,2,2] => 11010 => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,1] => 11001 => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,4] => 11000 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,4] => 11000 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,1] => 11001 => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,1] => 11001 => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,4] => 11000 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,2,1,1] => 11011 => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,3,1] => 11001 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,2,2] => 11010 => 2
Description
The number of descents of a binary word.
Matching statistic: St000665
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 0
[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[2,1] => [2]
 => [[1,2]]
 => [1,2] => 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
Description
The number of rafts of a permutation. Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St000834
(load all 3 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 0
[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[2,1] => [2]
 => [[1,2]]
 => [1,2] => 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
Description
The number of right outer peaks of a permutation. A right 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 $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000884
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => [1] => [1] => 0
[1,2] => {{1},{2}}
 => [1,2] => [1,2] => 0
[2,1] => {{1,2}}
 => [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => [2,3,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[2,4,3,1] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,1,3,2] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 1
[4,2,1,3] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,4,2,5,3] => 2
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001928
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => [1] => [1] => 0
[1,2] => {{1},{2}}
 => [1,2] => [1,2] => 0
[2,1] => {{1,2}}
 => [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => [2,3,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[2,4,3,1] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,1,3,2] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 1
[4,2,1,3] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,4,2,5,3] => 2
Description
The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000470
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => [1] => [1] => 1 = 0 + 1
[1,2] => {{1},{2}}
 => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => {{1,2}}
 => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => {{1,2,3}}
 => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[2,4,3,1] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 2 = 1 + 1
[3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[3,2,4,1] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 2 = 1 + 1
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[3,4,2,1] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[4,1,3,2] => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 2 = 1 + 1
[4,2,1,3] => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 2 = 1 + 1
[4,2,3,1] => {{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[4,3,1,2] => {{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 3 = 2 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,4,5,2,3] => 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,3,5,2,4] => 2 = 1 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000668The least common multiple of the parts of the partition. St000035The number of left outer peaks of a permutation. St000245The number of ascents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000325The width of the tree associated to 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.