Your data matches 487 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000144
(load all 72 compositions to match this statistic)
Mapping
St000144: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1
[1,0,1,0]
 => 2
[1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => 4
[1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => 5
[1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 6
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000998
(load all 70 compositions to match this statistic)
Mapping
St000998: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001240
(load all 70 compositions to match this statistic)
Mapping
St001240: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St000967
(load all 49 compositions to match this statistic)
Mapping
St000967: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 3 = 1 + 2
[1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,0,1,0,1,0]
 => 6 = 4 + 2
[1,0,1,0,1,1,0,0]
 => 6 = 4 + 2
[1,0,1,1,0,0,1,0]
 => 6 = 4 + 2
[1,0,1,1,1,0,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,0,1,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,1,0,0,0,1,0]
 => 6 = 4 + 2
[1,1,1,1,0,0,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,1,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,1,0,0,1,0]
 => 7 = 5 + 2
[1,0,1,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,0,1,1,0,0,1,0,1,0]
 => 7 = 5 + 2
[1,0,1,1,0,0,1,1,0,0]
 => 7 = 5 + 2
[1,0,1,1,1,0,0,0,1,0]
 => 7 = 5 + 2
[1,0,1,1,1,1,0,0,0,0]
 => 7 = 5 + 2
[1,1,0,0,1,0,1,0,1,0]
 => 7 = 5 + 2
[1,1,0,0,1,0,1,1,0,0]
 => 7 = 5 + 2
[1,1,0,0,1,1,0,0,1,0]
 => 7 = 5 + 2
[1,1,0,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,1,1,0,0,0,1,0,1,0]
 => 7 = 5 + 2
[1,1,1,0,0,0,1,1,0,0]
 => 7 = 5 + 2
[1,1,1,1,0,0,0,0,1,0]
 => 7 = 5 + 2
[1,1,1,1,1,0,0,0,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 8 = 6 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 8 = 6 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 8 = 6 + 2
Description
The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.
Matching statistic: St001012
(load all 37 compositions to match this statistic)
Mapping
Mp00222: Dyck paths ā€”peaks-to-valleysāŸ¶ Dyck paths
St001012: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 7 = 6 + 1
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001179
(load all 37 compositions to match this statistic)
Mapping
Mp00222: Dyck paths ā€”peaks-to-valleysāŸ¶ Dyck paths
St001179: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 7 = 6 + 1
Description
Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.
Matching statistic: St001237
(load all 121 compositions to match this statistic)
Mapping
Mp00327: Dyck paths ā€”inverse Kreweras complementāŸ¶ Dyck paths
St001237: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 7 = 6 + 1
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St000031
(load all 3 compositions to match this statistic)
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00223: Permutations ā€”runsortāŸ¶ Permutations
St000031: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [1,2] => 2
[1,1,0,0]
 => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => 4
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => 5
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => 5
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => 5
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => 5
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,3,4,5] => 5
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => 5
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,2,3,4,5,6] => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [1,2,3,4,5,6] => 6
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000050
(load all 28 compositions to match this statistic)
Mapping
Mp00132: Dyck paths ā€”switch returns and last double riseāŸ¶ Dyck paths
Mp00029: Dyck paths ā€”to binary tree: left tree, up step, right tree, down stepāŸ¶ Binary trees
St000050: Binary trees āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [.,.]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [[.,.],.]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => [.,[.,.]]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 3
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 4
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 5
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 5
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[[.,[[.,.],.]],.],.],.]
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => 6
Description
The depth or height of a binary tree. The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices. The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root. See [1] and [2] for this terminology. The depth (or height) of a tree $T$ can be recursively defined: $\operatorname{depth}(T) = 0$ if $T$ is empty and $$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$$ if $T$ is nonempty with left and right subtrees $L$ and $R$, respectively. The upper and lower bounds on the depth of a binary tree $T$ of size $n$ are $log_2(n) \leq \operatorname{depth}(T) \leq n$.
Matching statistic: St000056
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00223: Permutations ā€”runsortāŸ¶ Permutations
St000056: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [1,2] => 2
[1,1,0,0]
 => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => 4
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => 5
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => 5
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => 5
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => 5
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,3,4,5] => 5
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => 5
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,2,3,4,5,6] => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,2,3,4,5,6] => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [1,2,3,4,5,6] => 6
Description
The decomposition (or block) number of a permutation. For $\pi \in \mathcal{S}_n$, this is given by $$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].
The following 477 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000062The length of the longest increasing subsequence of the permutation. St000153The number of adjacent cycles of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000636The hull number of a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000991The number of right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001096The size of the overlap set of a permutation. St001176The size of a partition minus its first part. St001298The number of repeated entries in the Lehmer code of a permutation. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001405The number of bonds in a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001958The degree of the polynomial interpolating the values of a permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000203The number of external nodes of a binary tree. St000211The rank of the set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000676The number of odd rises of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000703The number of deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000839The largest opener of a set partition. St000874The position of the last double rise in a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to 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 Dyck path as follows: St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001322The size of a minimal independent dominating set in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001571The Cartan determinant of the integer partition. St001616The number of neutral elements in a lattice. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001691The number of kings in a graph. St001717The largest size of an interval in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001780The order of promotion on the set of standard tableaux of given shape. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000145The Dyson rank of a partition. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000228The size of a partition. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000459The hook length of the base cell of a partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000505The biggest entry in the block containing the 1. St000546The number of global descents of a permutation. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000662The staircase size of the code of a permutation. St000675The number of centered multitunnels of a Dyck path. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001759The Rajchgot index of a permutation. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000439The position of the first down step of a Dyck path. St000806The semiperimeter of the associated bargraph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000064The number of one-box pattern of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000060The greater neighbor of the maximum. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001052The length of the exterior of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000673The number of non-fixed points of a permutation. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000354The number of recoils of a permutation. St000392The length of the longest run of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000643The size of the largest orbit of antichains under Panyushev complementation. St000653The last descent of a permutation. St000657The smallest part of an integer composition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000744The length of the path to the largest entry in a standard Young tableau. St000794The mak of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000877The depth of the binary word interpreted as a path. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000691The number of changes of a binary word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001651The Frankl number of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000395The sum of the heights of the peaks of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000336The leg major index of a standard tableau. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001917The order of toric promotion on the set of labellings of a graph. St001645The pebbling number of a connected graph. St000890The number of nonzero entries in an alternating sign matrix. St001725The harmonious chromatic number of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000161The sum of the sizes of the right subtrees of a binary tree. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000071The number of maximal chains in a poset. St000451The length of the longest pattern of the form k 1 2. St000374The number of exclusive right-to-left minima of a permutation. St000651The maximal size of a rise in a permutation. St000356The number of occurrences of the pattern 13-2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000463The number of admissible inversions of a permutation. St000845The maximal number of elements covered by an element in a poset. St000883The number of longest increasing subsequences of a permutation. St000924The number of topologically connected components of a perfect matching. St000911The number of maximal antichains of maximal size in a poset. St000210Minimum over maximum difference of elements in cycles. St001727The number of invisible inversions of a permutation. St000051The size of the left subtree of a binary tree. St000134The size of the orbit of an alternating sign matrix under gyration. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000443The number of long tunnels of a Dyck path. St000652The maximal difference between successive positions of a permutation. St000831The number of indices that are either descents or recoils. St001115The number of even descents of a permutation. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001304The number of maximally independent sets of vertices of a graph. St001480The number of simple summands of the module J^2/J^3. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001963The tree-depth of a graph. St000235The number of indices that are not cyclical small weak excedances. St000795The mad of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001481The minimal height of a peak of a Dyck path. St001684The reduced word complexity of a permutation. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001948The number of augmented double ascents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001668The number of points of the poset minus the width of the poset. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000197The number of entries equal to positive one in the alternating sign matrix. St000487The length of the shortest cycle of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001432The order dimension of the partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001742The difference of the maximal and the minimal degree in a graph. St001935The number of ascents in a parking function. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St000522The number of 1-protected nodes of a rooted tree. St000044The number of vertices of the unicellular map given by a perfect matching. St000454The largest eigenvalue of a graph if it is integral. St000521The number of distinct subtrees of an ordered tree. St000568The hook number of a binary tree. St001114The number of odd descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000075The orbit size of a standard tableau under promotion. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000094The depth of an ordered tree. St001469The holeyness of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001822The number of alignments of a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000632The jump number of the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001820The size of the image of the pop stack sorting operator. St000259The diameter of a connected graph. St000527The width of the poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001812The biclique partition number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001555The order of a signed permutation. St001782The order of rowmotion on the set of order ideals of a poset.