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Your data matches 537 different statistics following compositions of up to 3 maps.
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Matching statistic: St000405
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(load all 9 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
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 0
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => 0
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 0
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> [2,1,3] => 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [2,1,3] => 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [3,1,2] => 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [3,1,2] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => 0
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> [1,2,3,4,5] => 0
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> [1,2,3,4,5] => 0
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> [3,1,2,4,5] => 0
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> [3,1,2,4,5] => 0
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> [3,1,2,4,5] => 0
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => 0
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => 0
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 0
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 0
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> [3,1,2,4] => 0
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> [3,1,2,4] => 0
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => 0
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => 0
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => 0
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => 0
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => 0
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => 0
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> [3,4,1,2] => 0
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => 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: St001086
(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
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 0
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => 0
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => 0
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> [2,1,3] => 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [2,1,3] => 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [3,1,2] => 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [3,1,2] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => 0
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> [1,2,3,4,5] => 0
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> [1,2,3,4,5] => 0
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> [3,1,2,4,5] => 0
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> [3,1,2,4,5] => 0
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> [3,1,2,4,5] => 0
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => 0
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => 0
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => 0
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => 0
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => 0
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> [3,1,2,4] => 0
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> [3,1,2,4] => 0
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => 0
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => 0
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => 0
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => 0
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => 0
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => 0
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> [3,4,1,2] => 0
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => 0
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000068
(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
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000069
(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
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St001633
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
Description
The number of maximal chains in a poset.
Matching statistic: St000527
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000909
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 0 + 2
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 0 + 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 0 + 2
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000143
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 0
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000146
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[5,0,0],[4,0],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[4,0,0,0],[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[4,0,0,0],[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 0
[[3,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,0,0],[3,0],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[3,1,0,0],[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[3,1,0,0],[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[3,1,0,0],[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 0
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 0
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 0
Description
The Andrews-Garvan crank of a partition.
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$
c(\pi) =
\begin{cases}
l(\pi) &\text{if \(\omega(\pi)=0\)}\\
\mu(\pi) - \omega(\pi) &\text{otherwise}.
\end{cases}
$$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
The following 527 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000150The floored half-sum of the multiplicities 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. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000257The number of distinct parts of a partition that occur at least twice. St000268The number of strongly connected orientations of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000313The number of degree 2 vertices of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000347The inversion sum of a binary word. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000455The second largest eigenvalue of a graph if it is integral. St000474Dyson's crank of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000552The number of cut vertices of a graph. St000629The defect of a binary word. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000699The toughness times the least common multiple of 1,. St000731The number of double exceedences of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000948The chromatic discriminant of a graph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001119The length of a shortest maximal path in a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. 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. 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. 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. St001350Half of the Albertson index of a graph. St001351The Albertson index 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001374The Padmakar-Ivan index of a graph. St001395The number of strictly unfriendly partitions of a graph. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001485The modular major index of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001625The Möbius invariant of a lattice. St001626The number of maximal proper sublattices of a lattice. St001638The book thickness of a graph. 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. St001696The natural major index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of 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. St001764The number of non-convex subsets of vertices in a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. 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). 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. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St001970The signature of a graph. St000081The number of edges of a graph. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000263The Szeged index of a graph. St000265The Wiener index 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. St000271The chromatic index of a graph. St000272The treewidth of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts 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. St000993The multiplicity of the largest part of an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001120The length of a longest path in a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001214The aft of an integer partition. St001270The bandwidth of a graph. St001272The number of graphs with the same degree sequence. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. 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). St001479The number of bridges of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001512The minimum rank of a graph. St001527The cyclic permutation representation number of an integer partition. St001546The number of monomials in the Tutte polynomial of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St001644The dimension of a 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. St001694The number of maximal dissociation sets in a graph. 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. 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. St001743The discrepancy of a graph. St001754The number of tolerances of a finite lattice. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001792The arboricity of a graph. St001820The size of the image of the pop stack sorting operator. St001826The maximal number of leaves on a vertex of a graph. St001833The number of linear intervals in a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001881The number of factors of a lattice as a Cartesian product of lattices. St001933The largest multiplicity of a part in an integer partition. 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. St001957The number of Hasse diagrams with a given underlying undirected graph. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000312The number of leaves in a graph. St000346The number of coarsenings of a partition. St000350The sum of the vertex degrees of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000422The energy of a graph, if it is integral. St000453The number of distinct Laplacian eigenvalues of a graph. St000456The monochromatic index of a connected graph. St000465The first Zagreb index of a graph. St000468The Hosoya index of a graph. St000482The (zero)-forcing number of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000549The number of odd partial sums of an integer partition. St000571The F-index (or forgotten topological index) of a graph. St000636The hull number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000759The smallest missing part in an integer partition. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. 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. 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. St000915The Ore degree of a graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001093The detour number of a graph. 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. St001116The game chromatic number of a graph. 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 $eA$. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001368The number of vertices of maximal degree in a graph. St001432The order dimension of the partition. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001619The number of non-isomorphic sublattices of a lattice. St001620The number of sublattices of a lattice. St001654The monophonic hull number of a graph. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001692The number of vertices with higher degree than the average degree in a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000452The number of distinct eigenvalues of a graph. St000548The number of different non-empty partial sums of an integer partition. St000722The number of different neighbourhoods in a graph. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001303The number of dominating sets of vertices of a graph. St001746The coalition number of a graph. St000264The girth of a graph, which is not a tree. St000511The number of invariant subsets when acting with a permutation of given cycle type. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001344The neighbouring number of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. 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. St000210Minimum over maximum difference of elements in cycles. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000352The Elizalde-Pak rank of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing 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 $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000486The number of cycles of length at least 3 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. St000546The number of global descents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises 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. St000709The number of occurrences of 14-2-3 or 14-3-2. 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. 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. 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. St000804The number of occurrences of the vincular pattern |123 in a permutation. 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. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000962The 3-shifted major index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001434The number of negative sum pairs of a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001536The number of cyclic misalignments 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. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in 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. 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. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001947The number of ties in a parking function. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000570The Edelman-Greene number 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000990The first ascent of a permutation. St001162The minimum jump 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001468The smallest fixpoint of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000842The breadth of a permutation. St001260The permanent of an alternating sign matrix. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001397Number of pairs of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001902The number of potential covers of a poset. 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. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. 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. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. 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. St001575The minimal number of edges to add or remove to make a graph edge 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. 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. St001783The number of odd automorphisms of a graph. St001827The number of two-component spanning forests of a graph. St001828The Euler characteristic of a graph. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001117The game chromatic index of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001518The number of graphs with the same ordinary spectrum as the given graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000311The number of vertices of odd degree in a graph. St000343The number of spanning subgraphs of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St001642The Prague dimension of a graph. St001771The number of occurrences of the signed pattern 1-2 in 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. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000479The Ramsey number of a graph.
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