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Your data matches 962 different statistics following compositions of up to 3 maps.
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Matching statistic: St000080
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
Description
The rank of the poset.
Matching statistic: St000218
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000356
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000405
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000534
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of 2-rises of a permutation.
A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]].
Matching statistic: St000632
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000845
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> 0
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000871
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [1] => 0
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => 0
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
The following 952 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St001902The number of potential covers of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000189The number of elements in the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001268The size of the largest ordinal summand in the poset. St001343The dimension of the reduced incidence algebra of a poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St001779The order of promotion on the set of linear extensions of a poset. St001813The product of the sizes of the principal order filters in a poset. St001815The number of order preserving surjections from a poset to a total order. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000180The number of chains of a poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001909The number of interval-closed sets of a poset. St001709The number of homomorphisms to the three element chain of a poset. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000081The number of edges of a graph. St000095The number of triangles of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000296The length of the symmetric border of a binary word. St000302The determinant of the distance matrix of a connected graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000352The Elizalde-Pak rank of a permutation. St000359The number of occurrences of the pattern 23-1. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000403The Szeged index minus the Wiener index of a graph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000422The energy of a graph, if it is integral. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000454The largest eigenvalue of a graph if it is integral. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000463The number of admissible inversions of a permutation. St000465The first Zagreb index of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000546The number of global descents of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000571The F-index (or forgotten topological index) of a graph. St000629The defect of a binary word. St000637The length of the longest cycle in a graph. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000703The number of deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000731The number of double exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000778The metric dimension of a graph. St000825The sum of the major and the inverse major index of a permutation. St000828The spearman's rho of a permutation and the identity permutation. St000884The number of isolated descents of a permutation. St000915The Ore degree of a graph. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000995The largest even part of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001248Sum of the even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001349The number of different graphs obtained from the given graph by removing an edge. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001374The Padmakar-Ivan index of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001394The genus of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001541The Gini index of an integer partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001638The book thickness of a graph. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001649The length of a longest trail in a graph. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. St001696The natural major index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001764The number of non-convex subsets of vertices in a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000007The number of saliances of the permutation. St000010The length of the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000054The first entry of the permutation. St000086The number of subgraphs. St000087The number of induced subgraphs. St000088The row sums of the character table of the symmetric group. St000093The cardinality of a maximal independent set of vertices of a graph. St000096The number of spanning trees of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000137The Grundy value of an integer partition. St000146The Andrews-Garvan crank of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000228The size of a partition. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000268The number of strongly connected orientations of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000271The chromatic index of a graph. St000273The domination number of a graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000343The number of spanning subgraphs of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000349The number of different adjacency matrices of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000474Dyson's crank of a partition. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000517The Kreweras number of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000618The number of self-evacuating tableaux of given shape. St000636The hull number of a graph. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000862The number of parts of the shifted shape of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000883The number of longest increasing subsequences of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St000948The chromatic discriminant of a graph. St000972The composition number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001029The size of the core of a graph. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001127The sum of the squares of the parts of a partition. St001129The product of the squares of the parts of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001366The maximal multiplicity of a degree of a vertex of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001386The number of prime labellings of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001441The number of non-empty connected induced subgraphs of a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001463The number of distinct columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001546The number of monomials in the Tutte polynomial of a graph. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001679The number of subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001691The number of kings in a graph. St001694The number of maximal dissociation sets in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001716The 1-improper chromatic number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001746The coalition number of a graph. St001754The number of tolerances of a finite lattice. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001763The Hurwitz number of an integer partition. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001802The number of endomorphisms of a graph. St001820The size of the image of the pop stack sorting operator. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001833The number of linear intervals in a lattice. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001883The mutual visibility number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000759The smallest missing part in an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001619The number of non-isomorphic sublattices of a lattice. St001620The number of sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001706The number of closed sets in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001762The number of convex subsets of vertices in a graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000670The reversal length of a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000807The sum of the heights of the valleys of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001375The pancake length of a permutation. St001377The major index minus the number of inversions of a permutation. St001379The number of inversions plus the major index of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001513The number of nested exceedences of a permutation. St001519The pinnacle sum of a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001671Haglund's hag of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001715The number of non-records in a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001911A descent variant minus the number of inversions. St001928The number of non-overlapping descents in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000040The number of regions of the inversion arrangement of a permutation. St000058The order of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000124The cardinality of the preimage of the Simion-Schmidt map. St000255The number of reduced Kogan faces with the permutation as type. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001285The number of primes in the column sums of the two line notation of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001468The smallest fixpoint of a permutation. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001850The number of Hecke atoms of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000065The number of entries equal to -1 in an alternating sign matrix. St001947The number of ties in a parking function. St000074The number of special entries. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000055The inversion sum of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000616The inversion index of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001470The cyclic holeyness of a permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001874Lusztig's a-function for the symmetric group. St000020The rank of the permutation. St000037The sign of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000325The width of the tree associated to a permutation. St000638The number of up-down runs of a permutation. St000737The last entry on the main diagonal of a semistandard tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000842The breadth of a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000929The constant term of the character polynomial of an integer partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000170The trace of a semistandard tableau. St000402Half the size of the symmetry class of a permutation. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000847The number of standard Young tableaux whose descent set is the binary word. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St001313The number of Dyck paths above the lattice path given by a binary word. St001838The number of nonempty primitive factors of a binary word. St000226The convexity of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000673The number of non-fixed points of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000830The total displacement of a permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index 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. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001114The number of odd descents of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000690The size of the conjugacy class of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001344The neighbouring number of a permutation. St000961The shifted major index of a permutation. St000570The Edelman-Greene number of a permutation. St000216The absolute length of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001388The number of non-attacking neighbors of a permutation. St001731The factorization defect of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. 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 $K[x]/(x^n)$. St001220The width of a permutation. St000102The charge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001684The reduced word complexity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St000958The number of Bruhat factorizations of a permutation. St001260The permanent of an alternating sign matrix. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001890The maximum magnitude of the Möbius function of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000635The number of strictly order preserving maps of a poset into itself. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001410The minimal entry of a semistandard tableau. 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. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000177The number of free tiles in the pattern. St000178Number of free entries. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.
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