searching the database
Your data matches 212 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: St001230
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 0
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001465
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000214
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => 1
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000441
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,5,4,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [3,1,5,4,2] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [3,2,1,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000665
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,5,4,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [3,1,5,4,2] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [3,2,1,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1
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: St001061
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St001061: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
Description
The number of indices that are both descents and recoils of a permutation.
Matching statistic: St000502
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 1
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000150
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 89%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 89%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1]
=> 0
[1,1,0,0,1,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1}
[1,1,0,1,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [3,3]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [3,3]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [5]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [5]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [4,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [4,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [4]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [4]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [3]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [2,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [2,1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [2]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [2]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [2]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,2,2,2}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,2,2,2}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,2,2,2}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [5,5]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,2,3}
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St001466
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,5,3,4,2,6] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,7,5,4,3,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,7,4,5,3,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,7,4,3,6,5,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7,3,5,4,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,6,4,3,5,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,6,3,4,5,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,5,3,4,7,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,5,3,4,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,3,5,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,5,3,4,2,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of transpositions swapping cyclically adjacent numbers in a permutation.
Put differently, this is the number of adjacent two-cycles in the chord diagram of a permutation.
The following 202 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St000661The number of rises of length 3 of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000534The number of 2-rises of a permutation. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000120The number of left tunnels of a Dyck path. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000475The number of parts equal to 1 in a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000659The number of rises of length at least 2 of a Dyck path. St000944The 3-degree of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000650The number of 3-rises of a permutation. St000872The number of very big descents of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000648The number of 2-excedences of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St001651The Frankl number of a lattice. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001403The number of vertical separators in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000241The number of cyclical small excedances. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001948The number of augmented double ascents of a permutation. St000137The Grundy value of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001961The sum of the greatest common divisors of all pairs of parts. St001820The size of the image of the pop stack sorting operator. St001095The number of non-isomorphic posets with precisely one further covering relation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000781The number of proper colouring schemes of a Ferrers diagram. St000993The multiplicity of the largest part of an integer partition. St001383The BG-rank of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000366The number of double descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000022The number of fixed points of a permutation. St001846The number of elements which do not have a complement in the lattice. St000908The length of the shortest maximal antichain in a poset. St000741The Colin de Verdière graph invariant. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001810The number of fixed points of a permutation smaller than its largest moved point. St001050The number of terminal closers of a set partition. St001621The number of atoms of a lattice.
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!