searching the database
Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000994
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000994: 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] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 001 => 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 001 => 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[] => [] => [] => ? => ? = 0
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 001 => 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 001 => 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 011 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[] => [] => [] => ? => ? = 0
Description
The number of runs of ones in a binary word.
Matching statistic: St001280
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => {{1}}
=> [1]
=> 0
[1,2] => [1,2] => {{1},{2}}
=> [1,1]
=> 0
[2,1] => [1,2] => {{1},{2}}
=> [1,1]
=> 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> [2,1]
=> 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> 1
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> 1
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
=> ?
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => {{1},{2,7},{3},{4,6},{5},{8}}
=> ?
=> ? = 2
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
=> ?
=> ? = 2
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => {{1},{2,7},{3},{4,6},{5},{8}}
=> ?
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
=> ?
=> ? = 2
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
=> ?
=> ? = 2
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => {{1},{2,7},{3},{4,5},{6},{8}}
=> ?
=> ? = 2
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ?
=> ? = 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => {{1},{2,6},{3,8},{4},{5},{7}}
=> ?
=> ? = 2
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => {{1},{2,6},{3},{4,7},{5},{8}}
=> ?
=> ? = 2
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
=> ?
=> ? = 2
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => {{1},{2,6},{3},{4,7},{5},{8}}
=> ?
=> ? = 2
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[8,6,5,7,2,3,4,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[8,6,5,4,3,2,7,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,5,6,4,3,2,7,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[8,6,4,5,3,2,7,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,5,4,6,3,2,7,1] => [1,8,2,5,3,4,6,7] => {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? = 2
[8,6,5,3,4,2,7,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,6,4,3,5,2,7,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
[8,6,3,4,5,2,7,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000703
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 83%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => {{1}}
=> [1] => 0
[1,2] => [1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => [1,2] => {{1},{2}}
=> [1,2] => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 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},{3},{4}}
=> [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[3,2,1,4] => [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
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[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},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => {{1},{2,4,6},{3,7},{5}}
=> [1,4,7,6,5,2,3] => ? = 2
[4,1,3,7,5,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,3,7,6,5,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,5,7,3,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,5,7,6,3,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,1,7,3,5,6] => [1,4,7,6,5,3,2] => {{1},{2,4,6},{3,7},{5}}
=> [1,4,7,6,5,2,3] => ? = 2
[4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,3,7,6,5,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,5,7,3,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,5,7,6,3,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,1,7,5,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,1,7,6,5,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,2,7,5,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,2,7,6,5,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,1,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,2,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,6,1,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,6,2,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => {{1},{2,5},{3,7},{4,6}}
=> [1,5,7,6,2,4,3] => ? = 3
[5,1,2,7,4,3,6] => [1,5,4,7,6,3,2] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => ? = 2
[5,1,3,4,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,3,6,7,4,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,3,7,2,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,3,7,2,6,4] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,4,3,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,4,6,7,3,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,4,7,2,3,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,4,7,2,6,3] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,2,1,3,7,4,6] => [1,5,7,6,4,3,2] => {{1},{2,5},{3,7},{4,6}}
=> [1,5,7,6,2,4,3] => ? = 3
[5,2,1,7,4,3,6] => [1,5,4,7,6,3,2] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => ? = 2
[5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,3,6,7,4,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,3,7,1,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,2,3,7,1,6,4] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,2,4,3,7,6,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,4,6,7,3,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,4,7,1,3,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,2,4,7,1,6,3] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,3,1,4,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,1,6,7,4,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,1,7,2,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,3,1,7,2,6,4] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,3,2,4,7,6,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,2,6,7,4,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,2,7,1,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,3,2,7,1,6,4] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,3,4,1,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,4,2,7,6,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,4,6,7,1,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,3,4,6,7,2,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000251
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 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] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[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] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 2
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [1,3,5,7,8,6,2,4] => ?
=> ? = 2
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [1,3,5,7,8,4,2,6] => ?
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [1,3,5,7,6,8,4,2] => ?
=> ? = 2
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [1,3,5,7,6,8,2,4] => ?
=> ? = 2
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [1,3,5,7,6,4,8,2] => ?
=> ? = 2
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [1,3,5,7,6,4,2,8] => ?
=> ? = 2
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 2
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [1,3,5,7,8,6,2,4] => ?
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [1,3,5,7,8,4,2,6] => ?
=> ? = 2
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [1,3,5,6,7,8,4,2] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [1,3,5,6,7,4,8,2] => ?
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [1,3,5,6,7,4,2,8] => ?
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 2
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [1,3,5,7,8,6,2,4] => ?
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [1,3,5,7,6,8,4,2] => ?
=> ? = 2
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [1,3,5,7,6,8,2,4] => ?
=> ? = 2
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [1,3,5,7,4,6,8,2] => ?
=> ? = 2
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
=> ? = 2
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [1,3,5,7,6,8,4,2] => ?
=> ? = 2
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [1,3,5,7,6,8,2,4] => ?
=> ? = 2
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [1,3,5,6,7,8,4,2] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [1,3,5,7,6,8,4,2] => ?
=> ? = 2
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [1,3,5,7,6,8,2,4] => ?
=> ? = 2
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [1,3,5,6,7,8,4,2] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => [1,3,5,6,7,8,4,2] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => [1,3,5,6,7,8,2,4] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => [1,3,5,6,4,7,8,2] => ?
=> ? = 2
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [1,3,5,6,4,7,2,8] => ?
=> ? = 2
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => [1,3,4,5,6,7,2,8] => ?
=> ? = 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => [1,4,5,6,7,2,8,3] => ?
=> ? = 2
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 2
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => [1,3,5,7,6,8,4,2] => ?
=> ? = 2
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => [1,3,5,7,6,4,8,2] => ?
=> ? = 2
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => [1,3,5,7,6,2,4,8] => ?
=> ? = 2
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => [1,3,5,7,8,6,4,2] => {{1},{2,3,5,8},{4,7},{6}}
=> ? = 2
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => [1,3,5,7,8,2,4,6] => ?
=> ? = 2
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => [1,3,5,6,7,8,4,2] => {{1},{2,3,5,7},{4,6,8}}
=> ? = 2
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => [1,3,5,6,7,4,8,2] => ?
=> ? = 2
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => [1,3,5,6,7,2,4,8] => ?
=> ? = 2
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [1,3,5,7,6,4,8,2] => ?
=> ? = 2
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => [1,3,5,7,4,6,8,2] => ?
=> ? = 2
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => [1,3,5,7,8,4,6,2] => ?
=> ? = 2
[8,6,5,7,2,3,4,1] => [1,8,2,6,3,5,4,7] => [1,3,5,7,6,4,8,2] => ?
=> ? = 2
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St001269
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => {{1}}
=> [1] => 0
[1,2] => [1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => [1,2] => {{1},{2}}
=> [1,2] => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 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},{3},{4}}
=> [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[3,2,1,4] => [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
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[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},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => {{1},{2,4,6},{3,7},{5}}
=> [1,4,7,6,5,2,3] => ? = 2
[4,1,3,6,5,7,2] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,3,7,5,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,3,7,6,5,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,5,6,3,7,2] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,5,7,3,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,1,5,7,6,3,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,1,6,3,7,5] => [1,4,6,7,5,3,2] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,1,7,3,5,6] => [1,4,7,6,5,3,2] => {{1},{2,4,6},{3,7},{5}}
=> [1,4,7,6,5,2,3] => ? = 2
[4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,3,7,6,5,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,5,6,3,7,1] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,5,7,3,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,2,5,7,6,3,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,1,6,5,7,2] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,3,1,7,5,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,1,7,6,5,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,2,6,5,7,1] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,3,2,7,5,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,2,7,6,5,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,6,1,7,2] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => {{1},{2,4,7},{3,6},{5}}
=> [1,4,6,7,5,3,2] => ? = 2
[4,3,5,7,1,6,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,2,6,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,6,1,2] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[4,3,5,7,6,2,1] => [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => {{1},{2,5},{3,6},{4,7}}
=> [1,5,6,7,2,3,4] => ? = 3
[5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => {{1},{2,5},{3,7},{4,6}}
=> [1,5,7,6,2,4,3] => ? = 3
[5,1,2,7,4,3,6] => [1,5,4,7,6,3,2] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => ? = 2
[5,1,2,7,6,4,3] => [1,5,6,4,7,3,2] => {{1},{2,5,7},{3,6},{4}}
=> [1,5,6,4,7,3,2] => ? = 2
[5,1,3,4,6,7,2] => [1,5,6,7,2,3,4] => {{1},{2,5},{3,6},{4,7}}
=> [1,5,6,7,2,3,4] => ? = 3
[5,1,3,4,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,3,6,7,4,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,3,7,2,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,3,7,2,6,4] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,4,3,6,7,2] => [1,5,6,7,2,3,4] => {{1},{2,5},{3,6},{4,7}}
=> [1,5,6,7,2,3,4] => ? = 3
[5,1,4,3,7,6,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,4,6,7,3,2] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,1,4,7,2,3,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,1,4,7,2,6,3] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[5,2,1,3,6,7,4] => [1,5,6,7,4,3,2] => {{1},{2,5},{3,6},{4,7}}
=> [1,5,6,7,2,3,4] => ? = 3
[5,2,1,3,7,4,6] => [1,5,7,6,4,3,2] => {{1},{2,5},{3,7},{4,6}}
=> [1,5,7,6,2,4,3] => ? = 3
[5,2,1,7,4,3,6] => [1,5,4,7,6,3,2] => {{1},{2,5,6},{3,4,7}}
=> [1,5,4,7,6,2,3] => ? = 2
[5,2,1,7,6,4,3] => [1,5,6,4,7,3,2] => {{1},{2,5,7},{3,6},{4}}
=> [1,5,6,4,7,3,2] => ? = 2
[5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => {{1},{2,5},{3,6},{4,7}}
=> [1,5,6,7,2,3,4] => ? = 3
[5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,3,6,7,4,1] => [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[5,2,3,7,1,4,6] => [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St000035
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[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] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => ? = 1
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => ? = 1
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 1
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => ? = 1
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => ? = 1
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 1
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 1
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 1
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 1
[1,2,6,3,4,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,5,6,4] => ? = 2
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 1
[1,2,6,3,5,7,4] => [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => ? = 2
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 1
[1,2,6,3,7,5,4] => [1,2,3,6,5,7,4] => [1,2,3,5,7,6,4] => ? = 1
[1,2,6,4,3,5,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 1
[1,2,6,4,3,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,5,6,4] => ? = 2
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 1
[1,2,6,4,5,7,3] => [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => ? = 2
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000834
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,2,6,3,4,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 1
[1,2,6,3,5,7,4] => [1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 1
[1,2,6,3,7,5,4] => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1
[1,2,6,4,3,5,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,2,6,4,3,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 1
[1,2,6,4,5,7,3] => [1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,6,4,7,3,5] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 1
[1,2,6,4,7,5,3] => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1
[1,2,6,5,3,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 1
[1,2,6,5,3,7,4] => [1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 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: St000353
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [3,1,2] => 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] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [3,1,2,4] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => [3,1,2,4] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 1
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 1
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => ? = 1
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 1
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 1
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => ? = 1
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 1
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 1
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 2
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 1
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 1
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000092The number of outer peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000640The rank of the largest boolean interval in a poset. St001624The breadth of a lattice. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. 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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!