Your data matches 108 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000431
(load all 6 compositions to match this statistic)
Mapping
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000431: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => {{1}}
 => [1] => 0
[[1,2]]
 => {{1,2}}
 => [2,1] => 0
[[1],[2]]
 => {{1},{2}}
 => [1,2] => 0
[[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 0
[[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 1
[[1,2],[3]]
 => {{1,2},{3}}
 => [2,1,3] => 1
[[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 0
[[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 2
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => [2,3,1,4] => 2
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 2
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 0
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 0
[[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 0
[[1,2,3,4,5,6,7]]
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => 0
Description
The number of occurrences of the pattern 213 or of the pattern 321 in a permutation.
Matching statistic: St001633
(load all 5 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [1,2] => ([(0,1)],2)
 => 0
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 0
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,2],[3]]
 => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2,3],[4]]
 => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000003
(load all 3 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1]
 => 1 = 0 + 1
[[1,2]]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7]
 => 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St001780
(load all 3 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1]
 => 1 = 0 + 1
[[1,2]]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7]
 => 1 = 0 + 1
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St001908
(load all 3 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001908: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1]
 => 1 = 0 + 1
[[1,2]]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7]
 => 1 = 0 + 1
Description
The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. For example, there are eight tableaux of shape $[3,2,1]$ with maximal entry $3$, but two of them have the same weight.
Matching statistic: St000057
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1]
 => [[1]]
 => 0
[[1,2]]
 => [1,2] => [1,1]
 => [[1],[2]]
 => 0
[[1],[2]]
 => [2,1] => [2]
 => [[1,2]]
 => 0
[[1,2,3]]
 => [1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 0
[[1,3],[2]]
 => [2,1,3] => [2,1]
 => [[1,2],[3]]
 => 1
[[1,2],[3]]
 => [3,1,2] => [2,1]
 => [[1,2],[3]]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [3]
 => [[1,2,3]]
 => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => [[1,2,3],[4]]
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => [[1,2,3,4]]
 => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5]
 => [[1,2,3,4,5]]
 => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6]
 => [[1,2,3,4,5,6]]
 => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7]
 => [[1,2,3,4,5,6,7]]
 => 0
Description
The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000220
(load all 2 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000425
(load all 6 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000433
(load all 7 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000433: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.
Matching statistic: St000457
(load all 23 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
Description
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000682The Grundy value of Welter's game on a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St000047The number of standard immaculate tableaux of a given shape. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001313The number of Dyck paths above the lattice path given by a binary word. St001595The number of standard Young tableaux of the skew partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000543The size of the conjugacy class of a binary word. St000293The number of inversions of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000529The number of permutations whose descent word is the given binary word. St000626The minimal period of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000290The major index of a binary word. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001377The major index minus the number of inversions of a permutation. St001485The modular major index of a binary word. St001910The height of the middle non-run of a Dyck path. St000075The orbit size of a standard tableau under promotion. St000100The number of linear extensions of a poset. St000548The number of different non-empty partial sums of an integer partition. 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. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St000530The number of permutations with the same descent word as the given permutation. St000222The number of alignments in the permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001742The difference of the maximal and the minimal degree in a graph. St000525The number of posets with the same zeta polynomial. St001246The maximal difference between two consecutive entries of a permutation. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St001510The number of self-evacuating linear extensions of a finite poset. St001760The number of prefix or suffix reversals needed to sort a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000045The number of linear extensions of a binary tree. St001811The Castelnuovo-Mumford regularity of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St000102The charge of a semistandard tableau. St001118The acyclic chromatic index of a graph. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001890The maximum magnitude of the Möbius function of a poset.