searching the database
Your data matches 113 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: St001212
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
St001212: 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]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St000035
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000035: 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]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 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: St000021
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 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: St000157
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000374
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: 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]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000659
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 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]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000834
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000834: 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]
=> [2,1] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,4,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,5,3,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [5,1,2,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,1,3,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [5,1,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000884
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 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].
Matching statistic: St000985
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ 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]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St000994
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000994: 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]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,4,3,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,4,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,4,5,3,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [2,5,3,4,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,4,2,5,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,4,2,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,5,2,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,4,2,3,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,5,4,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,1,4,3,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,4,1,5,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,5,1,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [2,3,5,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,5,2,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
The following 103 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001354The number of series nodes in the modular decomposition of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000142The number of even parts of a partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000245The number of ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000386The number of factors DDU in a Dyck path. St000390The number of runs of ones in a binary word. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000919The number of maximal left branches of a binary tree. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001657The number of twos in an integer partition. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000167The number of leaves of an ordered tree. St000201The number of leaf nodes in a binary tree. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000568The hook number of a binary tree. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001389The number of partitions of the same length below the given integer partition. St001674The number of vertices of the largest induced star graph in the graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000354The number of recoils of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000658The number of rises of length 2 of a Dyck path. St000829The Ulam distance of a permutation to the identity permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000647The number of big descents of a permutation. St001469The holeyness of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000099The number of valleys of a permutation, including the boundary. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001896The number of right descents of a signed permutations. St001330The hat guessing number of a graph. St001597The Frobenius rank of a skew partition. St000807The sum of the heights of the valleys of the associated bargraph. St001470The cyclic holeyness of a permutation. St001946The number of descents in a parking function. St000805The number of peaks of the associated bargraph. St001487The number of inner corners of a skew partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001624The breadth 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!