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Your data matches 431 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[2],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[3],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[4],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[3],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[4],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[3],[4],[5]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[2],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[2],[4]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[3],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[3],[4]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,3],[3],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,4],[3],[4]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[2],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[3],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[4],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[5],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[3],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[4],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[5],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[3],[4],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[3],[5],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[4],[5],[6]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2],[5]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[3],[5]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[4],[5]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[2],[5]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3],[5]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001198
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
Description
The 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 $eA$.
Matching statistic: St001206
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
Description
The 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 $eA$.
Matching statistic: St001256
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 2
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St001292
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 3 - 3
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 3 - 3
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 3 - 3
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 3 - 3
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 3
Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].
Matching statistic: St000542
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000955
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[4],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[3],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[4],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001194
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[4]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[4],[5]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[3],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[3],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[2],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[4],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[3],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[4],[5],[6]]
=> [3,2,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[3],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[4],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3],[5]]
=> [4,3,1,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
Description
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module
Matching statistic: St001239
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 100%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 3 - 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
The following 421 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001471The magnitude of a Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000056The decomposition (or block) number of a permutation. St000487The length of the shortest cycle 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. St000570The Edelman-Greene number of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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. St001162The minimum jump of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001820The size of the image of the pop stack sorting operator. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001881The number of factors of a lattice as a Cartesian product of lattices. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000210Minimum over maximum difference of elements in cycles. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000674The number of hills of a Dyck path. 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. 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. St000879The number of long braid edges in the graph of braid moves of a permutation. St000962The 3-shifted major index of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001381The fertility of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001513The number of nested exceedences 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. 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. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. 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. St001703The villainy of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. 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. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. 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. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000031The number of cycles in the cycle decomposition of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. 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. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000699The toughness times the least common multiple of 1,. 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. 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. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001260The permanent of an alternating sign matrix. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001722The number of minimal chains with small intervals between a binary word and the top element. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St001490The number of connected components of a skew partition. St000017The number of inversions of a standard tableau. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001371The length of the longest Yamanouchi prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001330The hat guessing number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001487The number of inner corners of a skew partition. St001890The maximum magnitude of the Möbius function 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. St001948The number of augmented double ascents of a permutation. St001488The number of corners of a skew partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000069The number of maximal elements of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000287The number of connected components of a graph. St000990The first ascent of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001871The number of triconnected components of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001430The number of positive entries in a signed permutation. St001520The number of strict 3-descents. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001868The number of alignments of type NE of a signed permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000567The sum of the products of all pairs of parts. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000116The major index of a semistandard tableau obtained by standardizing. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001651The Frankl number of a lattice. St001555The order of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000100The number of linear extensions of a poset. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000264The girth of a graph, which is not a tree. St000635The number of strictly order preserving maps of a poset into itself. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in 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. 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. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. 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. St000717The number of ordinal summands of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001864The number of excedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St000068The number of minimal elements in a poset. St000911The number of maximal antichains of maximal size in a poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001768The number of reduced words of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. 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 $K[x]/(x^n)$. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001851The number of Hecke atoms of a signed permutation. St001863The number of weak excedances of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001410The minimal entry of a semistandard tableau. St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St000307The number of rowmotion orbits of a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001623The number of doubly irreducible elements of a lattice. St001754The number of tolerances of a finite lattice. St000080The rank of the poset. St000084The number of subtrees. St000094The depth of an ordered tree. St000168The number of internal nodes of an ordered tree. St000189The number of elements in the poset. St000327The number of cover relations in a poset. St000328The maximum number of child nodes in a tree. St000413The number of ordered trees with the same underlying unordered tree. St000417The size of the automorphism group of the ordered tree. St000521The number of distinct subtrees of an ordered tree. St000679The pruning number of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001058The breadth of the ordered tree. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St001668The number of points of the poset minus the width of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001625The Möbius invariant of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000180The number of chains of a poset. St000400The path length of an ordered tree. St001909The number of interval-closed sets of a poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000529The number of permutations whose descent word is the given binary word. St000416The number of inequivalent increasing trees of an ordered tree. St000634The number of endomorphisms of a poset. St000410The tree factorial of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.
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