Your data matches 64 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000222
(load all 3 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000222: 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]]
 => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => 1
[[1,3],[2]]
 => [2,1,3] => [3,1,2] => 0
[[1,2],[3]]
 => [3,1,2] => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 2
[[1,3,4],[2]]
 => [2,1,3,4] => [4,3,1,2] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [4,2,1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [4,1,2,3] => 0
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0
Description
The number of alignments in the permutation.
Matching statistic: St001822
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed 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] => [2,1] => [2,1] => 0
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => [3,2,1] => 1
[[1,3],[2]]
 => [2,1,3] => [3,1,2] => [3,1,2] => 0
[[1,2],[3]]
 => [3,1,2] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[[1,3,4],[2]]
 => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of alignments of a signed permutation. An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]]. Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Matching statistic: St000432
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000432: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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] => [2,1] => 0
[[1,2,3]]
 => [1,2,3] => [1,3,2] => [3,1,2] => 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[1,2],[3]]
 => [3,1,2] => [3,1,2] => [2,3,1] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 2
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,4,3] => [1,4,2,3] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [4,1,3,2] => [4,2,3,1] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [2,4,1,3] => [1,3,4,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
Description
The number of occurrences of the pattern 231 or of the pattern 312 in a permutation.
Matching statistic: St001713
(load all 5 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Values
[[1]]
 => [[1]]
 => [[1]]
 => ? = 0 + 2
[[1,2]]
 => [[1,2]]
 => [[2,0],[1]]
 => 2 = 0 + 2
[[1],[2]]
 => [[1,2]]
 => [[2,0],[1]]
 => 2 = 0 + 2
[[1,2,3]]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3 = 1 + 2
[[1,3],[2]]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 2 = 0 + 2
[[1,2],[3]]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 3 = 1 + 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 2 = 0 + 2
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 2 + 2
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => ? = 1 + 2
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 2 + 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 2 + 2
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => ? = 1 + 2
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => ? = 2 + 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => ? = 0 + 2
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => ? = 2 + 2
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => ? = 2 + 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => ? = 0 + 2
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001879
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => ? = 0 + 1
[[1,2]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => ? = 0 + 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => ? = 0 + 1
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ? = 0 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ? = 0 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ? = 0 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ? = 0 + 1
Description
The 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.
Matching statistic: St001880
(load all 4 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => ? = 0 + 2
[[1,2]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => ? = 0 + 2
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => ? = 0 + 2
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ? = 0 + 2
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ? = 0 + 2
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ? = 0 + 2
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ? = 0 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000102
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [[1]]
 => [[1]]
 => 0
[[1,2]]
 => [1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => 0
[[1],[2]]
 => [2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => 0
[[1,2,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ? = 1
[[1,3],[2]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ? = 0
[[1,2],[3]]
 => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => ? = 1
[[1],[2],[3]]
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ? = 0
[[1,2,3,4]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ? = 2
[[1,3,4],[2]]
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ? = 1
[[1,2,4],[3]]
 => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => ? = 2
[[1,2,3],[4]]
 => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => ? = 2
[[1,3],[2,4]]
 => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => ? = 1
[[1,2],[3,4]]
 => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => ? = 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => ? = 0
[[1,3],[2],[4]]
 => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => ? = 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,3],[2,3,4],[3,4],[4]]
 => ? = 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,4],[2,3,4],[3,4],[4]]
 => ? = 0
Description
The charge of a semistandard tableau.
Matching statistic: St000477
Mapping
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1]
 => []
 => ? = 0
[[1,2]]
 => [2] => [2]
 => []
 => ? = 0
[[1],[2]]
 => [2] => [2]
 => []
 => ? = 0
[[1,2,3]]
 => [3] => [3]
 => []
 => ? = 1
[[1,3],[2]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2],[3]]
 => [2,1] => [2,1]
 => [1]
 => ? = 1
[[1],[2],[3]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2,3,4]]
 => [4] => [4]
 => []
 => ? = 2
[[1,3,4],[2]]
 => [4] => [4]
 => []
 => ? = 1
[[1,2,4],[3]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,2,3],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,3],[2,4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 1
[[1,2],[3,4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [4] => [4]
 => []
 => ? = 0
[[1,3],[2],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,2],[3],[4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [4] => [4]
 => []
 => ? = 0
Description
The weight of a partition according to Alladi.
Matching statistic: St000514
Mapping
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000514: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1]
 => []
 => ? = 0
[[1,2]]
 => [2] => [2]
 => []
 => ? = 0
[[1],[2]]
 => [2] => [2]
 => []
 => ? = 0
[[1,2,3]]
 => [3] => [3]
 => []
 => ? = 1
[[1,3],[2]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2],[3]]
 => [2,1] => [2,1]
 => [1]
 => ? = 1
[[1],[2],[3]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2,3,4]]
 => [4] => [4]
 => []
 => ? = 2
[[1,3,4],[2]]
 => [4] => [4]
 => []
 => ? = 1
[[1,2,4],[3]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,2,3],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,3],[2,4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 1
[[1,2],[3,4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [4] => [4]
 => []
 => ? = 0
[[1,3],[2],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,2],[3],[4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [4] => [4]
 => []
 => ? = 0
Description
The number of invariant simple graphs when acting with a permutation of given cycle type.
Matching statistic: St000515
Mapping
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1]
 => []
 => ? = 0
[[1,2]]
 => [2] => [2]
 => []
 => ? = 0
[[1],[2]]
 => [2] => [2]
 => []
 => ? = 0
[[1,2,3]]
 => [3] => [3]
 => []
 => ? = 1
[[1,3],[2]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2],[3]]
 => [2,1] => [2,1]
 => [1]
 => ? = 1
[[1],[2],[3]]
 => [3] => [3]
 => []
 => ? = 0
[[1,2,3,4]]
 => [4] => [4]
 => []
 => ? = 2
[[1,3,4],[2]]
 => [4] => [4]
 => []
 => ? = 1
[[1,2,4],[3]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,2,3],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,3],[2,4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 1
[[1,2],[3,4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [4] => [4]
 => []
 => ? = 0
[[1,3],[2],[4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2
[[1,2],[3],[4]]
 => [2,2] => [2,2]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [4] => [4]
 => []
 => ? = 0
Description
The number of invariant set partitions when acting with a permutation of given cycle type.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001060The distinguishing index of a graph. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St001890The maximum magnitude of the Möbius function of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000699The toughness times the least common multiple of 1,. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.