Your data matches 266 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000023
(load all 100 compositions to match this statistic)
Mapping
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St000353
(load all 66 compositions to match this statistic)
Mapping
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 0
Description
The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].
Matching statistic: St000092
(load all 66 compositions to match this statistic)
Mapping
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 1 = 0 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 1 = 0 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 1 = 0 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 1 = 0 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 1 = 0 + 1
[3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 1 = 0 + 1
[4,1,2,3] => 2 = 1 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 2 = 1 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 2 = 1 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 1 = 0 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 1 = 0 + 1
[1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => 1 = 0 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => 1 = 0 + 1
[1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => 1 = 0 + 1
[1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => 2 = 1 + 1
[1,4,3,2,5] => 2 = 1 + 1
[1,4,3,5,2] => 2 = 1 + 1
[1,4,5,2,3] => 2 = 1 + 1
[1,4,5,3,2] => 1 = 0 + 1
Description
The number of outer peaks of a permutation. An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 100 compositions to match this statistic)
Mapping
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 1 = 0 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 1 = 0 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 2 = 1 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 1 = 0 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => 2 = 1 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => 2 = 1 + 1
[1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => 2 = 1 + 1
[1,4,3,5,2] => 3 = 2 + 1
[1,4,5,2,3] => 2 = 1 + 1
[1,4,5,3,2] => 2 = 1 + 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000021
(load all 14 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000035
(load all 60 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
[1,4,5,3,2] => [1,2,4,3,5] => 1
Description
The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000196
(load all 12 compositions to match this statistic)
Mapping
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
 => 0
[2,1] => [[.,.],.]
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => 0
[1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [[.,.],[.,.]]
 => 1
[2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [[.,.],[.,.]]
 => 1
[3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
 => 0
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000291
(load all 17 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 0
[2,1,3] => 10 => 1
[2,3,1] => 01 => 0
[3,1,2] => 10 => 1
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 0
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 0
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 0
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 0
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 0
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 0
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 0
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 0
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 0
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 0
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 0
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 1
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 0
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 23 compositions to match this statistic)
Mapping
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
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 0
[2,3,1] => 01 => 1
[3,1,2] => 10 => 0
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 0
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 0
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 0
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 0
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 0
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 0
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000884
(load all 10 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => 1
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
The following 256 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000201The number of leaf nodes in a binary tree. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000386The number of factors DDU in a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000919The number of maximal left branches of a binary tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000054The first entry of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000345The number of refinements of a partition. St000527The width of the poset. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000159The number of distinct parts of the integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000356The number of occurrences of the pattern 13-2. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000387The matching number of a graph. St000536The pathwidth of a graph. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000409The number of pitchforks in a binary tree. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000912The number of maximal antichains in a poset. St000946The sum of the skew hook positions in a Dyck path. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001432The order dimension of the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001668The number of points of the poset minus the width of the poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001330The hat guessing number of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000284The Plancherel distribution on integer partitions. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001597The Frobenius rank of a skew partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001896The number of right descents of a signed permutations. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000307The number of rowmotion orbits of a poset. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001820The size of the image of the pop stack sorting operator. 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. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.