Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000186
(load all 5 compositions to match this statistic)
Mapping
St000186: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => 1
[[1,0],[1]]
 => 1
[[2,0],[0]]
 => 2
[[2,0],[1]]
 => 2
[[2,0],[2]]
 => 2
[[1,1],[1]]
 => 2
[[1,0,0],[0,0],[0]]
 => 1
[[1,0,0],[1,0],[0]]
 => 1
[[1,0,0],[1,0],[1]]
 => 1
[[3,0],[0]]
 => 3
[[3,0],[1]]
 => 3
[[3,0],[2]]
 => 3
[[3,0],[3]]
 => 3
[[2,1],[1]]
 => 3
[[2,1],[2]]
 => 3
[[2,0,0],[0,0],[0]]
 => 2
[[2,0,0],[1,0],[0]]
 => 2
[[2,0,0],[1,0],[1]]
 => 2
[[2,0,0],[2,0],[0]]
 => 2
[[2,0,0],[2,0],[1]]
 => 2
[[2,0,0],[2,0],[2]]
 => 2
[[1,1,0],[1,0],[0]]
 => 2
[[1,1,0],[1,0],[1]]
 => 2
[[1,1,0],[1,1],[1]]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => 1
[[4,0],[0]]
 => 4
[[4,0],[1]]
 => 4
[[4,0],[2]]
 => 4
[[4,0],[3]]
 => 4
[[4,0],[4]]
 => 4
[[3,1],[1]]
 => 4
[[3,1],[2]]
 => 4
[[3,1],[3]]
 => 4
[[2,2],[2]]
 => 4
[[3,0,0],[0,0],[0]]
 => 3
[[3,0,0],[1,0],[0]]
 => 3
[[3,0,0],[1,0],[1]]
 => 3
[[3,0,0],[2,0],[0]]
 => 3
[[3,0,0],[2,0],[1]]
 => 3
[[3,0,0],[2,0],[2]]
 => 3
[[3,0,0],[3,0],[0]]
 => 3
[[3,0,0],[3,0],[1]]
 => 3
[[3,0,0],[3,0],[2]]
 => 3
[[3,0,0],[3,0],[3]]
 => 3
[[2,1,0],[1,0],[0]]
 => 3
[[2,1,0],[1,0],[1]]
 => 3
[[2,1,0],[1,1],[1]]
 => 3
Description
The sum of the first row in a Gelfand-Tsetlin pattern.
Matching statistic: St000228
(load all 11 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000135
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000135: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2,1] => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1,3] => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [3,1,2] => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [2,1] => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [2,1] => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [2,1] => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [2,1,3,4] => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [3,1,2,4] => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [4,1,2,3] => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [3,4,1,2] => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [2,1,3] => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [2,1,3] => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 3
Description
The number of lucky cars of the parking function. A lucky car is a car that was able to park in its prefered spot. The generating function, $$ q\prod_{i=1}^{n-1} (i + (n-i+1)q) $$ was established in [1].
Matching statistic: St000189
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => ([],1)
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => ([(0,1)],2)
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => ([],1)
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => ([],1)
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => ([(1,2)],3)
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => ([(0,1)],2)
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => ([],2)
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => ([],2)
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => ([],1)
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => ([],1)
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => ([],1)
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 3
Description
The number of elements in the poset.
Matching statistic: St000197
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [[1]]
 => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [[1]]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [[0,1],[1,0]]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [[1]]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [[1]]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [[1]]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [[0,1],[1,0]]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [[0,1],[1,0]]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [[0,1],[1,0]]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [[1]]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [[1]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [[1]]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [[1]]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 3
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => 100 => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,1]
 => 110 => 2
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => 100 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 110 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 10 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 1000 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [2,1]
 => 1010 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [2,1]
 => 1010 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 1000 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => 1010 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => 1010 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => 100 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,1]
 => 110 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,1]
 => 110 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => 100 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,1]
 => 110 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => 100 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => 110 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => 110 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => 110 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => 10 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => 10 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => 10000 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [3,1]
 => 10010 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [2,2]
 => 1100 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [3,1]
 => 10010 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => 10000 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => 10010 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [2,2]
 => 1100 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => 10010 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => 1100 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => 1000 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,1,1]
 => 1110 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => 1000 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [2,1]
 => 1010 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => 1000 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => 1010 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => 1010 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [1,1,1]
 => 1110 => 3
Description
The number of inversions of a binary word.
Matching statistic: St000459
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [[1,2,2]]
 => [3]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [[3,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [[2,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [[1,3]]
 => [2]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [[2,3]]
 => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [[1,3]]
 => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [[4]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [[2,2,2,2]]
 => [4]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [[1,1,1,1]]
 => [4]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [[3,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [[2,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [[1,3,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [[2,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [[1,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [[1,1,3]]
 => [3]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [[2,3,3]]
 => [3]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [[1,3,3]]
 => [3]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [2,1]
 => 3
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [[1,2,2]]
 => [3]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [[3,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [[2,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [[1,3]]
 => [2]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [[2,3]]
 => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [[1,3]]
 => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [[4]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [[2,2,2,2]]
 => [4]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [[1,1,1,1]]
 => [4]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [[3,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [[2,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [[1,3,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [[2,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [[1,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [[1,1,3]]
 => [3]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [[2,3,3]]
 => [3]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [[1,3,3]]
 => [3]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [2,1]
 => 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
(load all 2 compositions to match this statistic)
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [[1,2,2]]
 => [3]
 => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [[1,1,2]]
 => [3]
 => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [[3,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [[2,3]]
 => [2]
 => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [[1,3]]
 => [2]
 => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [[2,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => [2]
 => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [[1,1]]
 => [2]
 => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [[2,3]]
 => [2]
 => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [[1,3]]
 => [2]
 => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => [2]
 => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [[4]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [[3]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [[2]]
 => [1]
 => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [[1]]
 => [1]
 => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [[2,2,2,2]]
 => [4]
 => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [[1,1,1,1]]
 => [4]
 => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [[1,2,2,2]]
 => [4]
 => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [[1,1,1,2]]
 => [4]
 => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [[1,1,2,2]]
 => [4]
 => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [[3,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [[2,3,3]]
 => [3]
 => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [[1,3,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [[2,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [[1,2,3]]
 => [3]
 => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [[1,1,3]]
 => [3]
 => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [[2,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [[1,2,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [[1,1,2]]
 => [3]
 => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [[1,1,1]]
 => [3]
 => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [[2,3,3]]
 => [3]
 => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [[1,3,3]]
 => [3]
 => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [2,1]
 => 3
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001004
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 2
[[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 2
[[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,2] => 2
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 1
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 3
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1,3] => [1,3,2] => 3
[[2,1],[2]]
 => [[1,1],[2]]
 => [3,1,2] => [1,2,3] => 3
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [1,2] => [1,2] => 2
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [2,1] => [1,2] => 2
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [2,1] => [1,2] => 2
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [2,1] => [1,2] => 2
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1] => [1] => 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1] => [1] => 1
[[4,0],[0]]
 => [[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[1]]
 => [[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[3]]
 => [[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[4,0],[4]]
 => [[1,1,1,1]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [2,1,3,4] => [1,3,4,2] => 4
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1,2,4] => [1,2,4,3] => 4
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [4,1,2,3] => [1,2,3,4] => 4
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [3,4,1,2] => [1,2,3,4] => 4
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [1,2,3] => [1,2,3] => 3
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [1,2,3] => [1,2,3] => 3
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1,3] => [1,3,2] => 3
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1,3] => [1,3,2] => 3
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1,3] => [1,3,2] => 3
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001430The number of positive entries in a signed permutation. St001622The number of join-irreducible elements of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001925The minimal number of zeros in a row of an alternating sign matrix. St001958The degree of the polynomial interpolating the values of a permutation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000625The sum of the minimal distances to a greater element. St000890The number of nonzero entries in an alternating sign matrix. St001074The number of inversions of the cyclic embedding of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000458The number of permutations obtained by switching adjacencies or successions. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000519The largest length of a factor maximising the subword complexity. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St001621The number of atoms of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.