Your data matches 102 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001627
Mapping
St001627: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 1
[3]
 => 2
[2,1]
 => 3
[1,1,1]
 => 4
[4]
 => 6
[3,1]
 => 11
[2,2]
 => 16
[2,1,1]
 => 23
[1,1,1,1]
 => 38
[5]
 => 21
[4,1]
 => 58
[3,2]
 => 98
[3,1,1]
 => 162
[2,2,1]
 => 230
[2,1,1,1]
 => 402
[1,1,1,1,1]
 => 728
[6]
 => 112
[5,1]
 => 407
[4,2]
 => 879
[4,1,1]
 => 1549
[3,3]
 => 1087
[3,2,1]
 => 2812
[3,1,1,1]
 => 5204
[2,2,2]
 => 4065
[2,2,1,1]
 => 7490
[2,1,1,1,1]
 => 14080
[1,1,1,1,1,1]
 => 26704
Description
The number of coloured connected graphs such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, [[oeis:A001349]], whereas the value on the partition $(1^n)$ is the number of labelled connected graphs [[oeis:A001187]].
Matching statistic: St001207
(load all 12 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? ∊ {3,4} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? ∊ {3,4} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? ∊ {6,11,16,23,38} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? ∊ {6,11,16,23,38} + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? ∊ {6,11,16,23,38} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? ∊ {6,11,16,23,38} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? ∊ {6,11,16,23,38} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? ∊ {21,58,98,162,230,402,728} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? ∊ {21,58,98,162,230,402,728} + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? ∊ {21,58,98,162,230,402,728} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? ∊ {21,58,98,162,230,402,728} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? ∊ {21,58,98,162,230,402,728} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? ∊ {21,58,98,162,230,402,728} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? ∊ {21,58,98,162,230,402,728} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704} + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000023
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St000162
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => [5,6,7,8,10,1,2,3,4,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => [6,7,8,9,10,12,1,2,3,4,5,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [4,6,7,8,10,1,2,3,5,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => [2,8,9,10,11,12,1,3,4,5,6,7] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => [7,8,9,10,11,12,14,1,2,3,4,5,6,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => [5,7,8,9,10,12,1,2,3,4,6,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [4,5,7,8,10,1,2,3,6,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => [2,7,9,10,11,12,1,3,4,5,6,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => [2,9,10,11,12,13,14,1,3,4,5,6,7,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of nontrivial cycles in the cycle decomposition of a permutation. This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000232
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 1
[2]
 => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => {{1,2,5},{3,4,6}}
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => {{1,3,4},{2,5,6}}
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => {{1,2,3,7},{4,5,6,8}}
 => ? ∊ {2,4}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => {{1,3,5},{2,4,6}}
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => {{1,3,4,5},{2,6,7,8}}
 => ? ∊ {2,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => {{1,2,3,4,9},{5,6,7,8,10}}
 => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => {{1,2,4,7},{3,5,6,8}}
 => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => {{1,2,5,6},{3,4,7,8}}
 => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => {{1,3,4,6},{2,5,7,8}}
 => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => {{1,3,4,5,6},{2,7,8,9,10}}
 => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => {{1,2,3,4,5,11},{6,7,8,9,10,12}}
 => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => {{1,2,3,5,9},{4,6,7,8,10}}
 => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => {{1,2,5,7},{3,4,6,8}}
 => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => {{1,3,4,7},{2,5,6,8}}
 => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => {{1,3,5,6},{2,4,7,8}}
 => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => {{1,3,4,5,7},{2,6,8,9,10}}
 => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => {{1,3,4,5,6,7},{2,8,9,10,11,12}}
 => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => {{1,2,3,4,5,6,13},{7,8,9,10,11,12,14}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => {{1,2,3,4,6,11},{5,7,8,9,10,12}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => {{1,2,3,6,9},{4,5,7,8,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => {{1,2,4,5,9},{3,6,7,8,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => {{1,2,3,7,8},{4,5,6,9,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => {{1,3,5,7},{2,4,6,8}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => {{1,3,4,5,8},{2,6,7,9,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => {{1,2,5,6,7},{3,4,8,9,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => {{1,3,4,6,7},{2,5,8,9,10}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => {{1,3,4,5,6,8},{2,7,9,10,11,12}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => {{1,3,4,5,6,7,8},{2,9,10,11,12,13,14}}
 => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of crossings of a set partition. This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000234
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of global ascents of a permutation. The global ascents are the integers $i$ such that $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$ Equivalently, by the pigeonhole principle, $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$ For $n > 1$ it can also be described as an occurrence of the mesh pattern $$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$ or equivalently $$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000243
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of cyclic valleys and cyclic peaks of a permutation. This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000353
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].
Matching statistic: St000472
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {2,3}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 4
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {2,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000486
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 14%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => ? ∊ {3,4}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => ? ∊ {3,4}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => [5,6,7,8,10,1,2,3,4,9] => ? ∊ {6,11,16,23,38}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => ? ∊ {6,11,16,23,38}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => ? ∊ {6,11,16,23,38}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => ? ∊ {6,11,16,23,38}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => ? ∊ {6,11,16,23,38}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => [6,7,8,9,10,12,1,2,3,4,5,11] => ? ∊ {21,58,98,162,230,402,728}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [4,6,7,8,10,1,2,3,5,9] => ? ∊ {21,58,98,162,230,402,728}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => ? ∊ {21,58,98,162,230,402,728}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => ? ∊ {21,58,98,162,230,402,728}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => ? ∊ {21,58,98,162,230,402,728}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => ? ∊ {21,58,98,162,230,402,728}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => [2,8,9,10,11,12,1,3,4,5,6,7] => ? ∊ {21,58,98,162,230,402,728}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => [7,8,9,10,11,12,14,1,2,3,4,5,6,13] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => [5,7,8,9,10,12,1,2,3,4,6,11] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [4,5,7,8,10,1,2,3,6,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => [2,7,9,10,11,12,1,3,4,5,6,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => [2,9,10,11,12,13,14,1,3,4,5,6,7,8] => ? ∊ {112,407,879,1087,1549,2812,4065,5204,7490,14080,26704}
Description
The number of cycles of length at least 3 of a permutation.
The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001625The Möbius invariant of a lattice. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001722The number of minimal chains with small intervals between a binary word and the top element. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000489The number of cycles of a permutation of length at most 3. St000502The number of successions of a set partitions. St000837The number of ascents of distance 2 of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001298The number of repeated entries in the Lehmer code of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001665The number of pure excedances of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St000037The sign of a permutation. St000490The intertwining number of a set partition. St000691The number of changes of a binary word. St000717The number of ordinal summands of a poset. St000824The sum of the number of descents and the number of recoils of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001424The number of distinct squares in a binary word. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000638The number of up-down runs 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. St001439The number of even weak deficiencies and of odd weak exceedences. St000632The jump number of the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.