Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St000242
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => 0
[[2,2]]
 => [1,2] => 0
[[1],[2]]
 => [2,1] => 0
[[1,3]]
 => [1,2] => 0
[[2,3]]
 => [1,2] => 0
[[3,3]]
 => [1,2] => 0
[[1],[3]]
 => [2,1] => 0
[[2],[3]]
 => [2,1] => 0
[[1,1,2]]
 => [1,2,3] => 0
[[1,2,2]]
 => [1,2,3] => 0
[[2,2,2]]
 => [1,2,3] => 0
[[1,1],[2]]
 => [3,1,2] => 3
[[1,2],[2]]
 => [2,1,3] => 1
[[1,4]]
 => [1,2] => 0
[[2,4]]
 => [1,2] => 0
[[3,4]]
 => [1,2] => 0
[[4,4]]
 => [1,2] => 0
[[1],[4]]
 => [2,1] => 0
[[2],[4]]
 => [2,1] => 0
[[3],[4]]
 => [2,1] => 0
[[1,1,3]]
 => [1,2,3] => 0
[[1,2,3]]
 => [1,2,3] => 0
[[1,3,3]]
 => [1,2,3] => 0
[[2,2,3]]
 => [1,2,3] => 0
[[2,3,3]]
 => [1,2,3] => 0
[[3,3,3]]
 => [1,2,3] => 0
[[1,1],[3]]
 => [3,1,2] => 3
[[1,2],[3]]
 => [3,1,2] => 3
[[1,3],[2]]
 => [2,1,3] => 1
[[1,3],[3]]
 => [2,1,3] => 1
[[2,2],[3]]
 => [3,1,2] => 3
[[2,3],[3]]
 => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => 1
[[1,1,1,2]]
 => [1,2,3,4] => 0
[[1,1,2,2]]
 => [1,2,3,4] => 0
[[1,2,2,2]]
 => [1,2,3,4] => 0
[[2,2,2,2]]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [4,1,2,3] => 4
[[1,1,2],[2]]
 => [3,1,2,4] => 3
[[1,2,2],[2]]
 => [2,1,3,4] => 1
[[1,1],[2,2]]
 => [3,4,1,2] => 4
[[1,5]]
 => [1,2] => 0
[[2,5]]
 => [1,2] => 0
[[3,5]]
 => [1,2] => 0
[[4,5]]
 => [1,2] => 0
[[5,5]]
 => [1,2] => 0
[[1],[5]]
 => [2,1] => 0
[[2],[5]]
 => [2,1] => 0
[[3],[5]]
 => [2,1] => 0
[[4],[5]]
 => [2,1] => 0
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000264
(load all 4 compositions to match this statistic)
Mapping
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[[2,2]]
 => [[2,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[[1,3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [[3,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [[2,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [[4,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [[3,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [[5,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001232
Mapping
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [[1,2]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2,2]]
 => [[2,2]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1],[2]]
 => [[1,2]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,3]]
 => [[1,3]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2,3]]
 => [[2,3]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3,3]]
 => [[3,3]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1],[3]]
 => [[1,3]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2],[3]]
 => [[2,3]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,2]]
 => [[1,1,2]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,3} + 1
[[1,2,2]]
 => [[1,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,3} + 1
[[2,2,2]]
 => [[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,3} + 1
[[1,1],[2]]
 => [[1,1,2]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,3} + 1
[[1,2],[2]]
 => [[1,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,1,3} + 1
[[1,4]]
 => [[1,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2,4]]
 => [[2,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3,4]]
 => [[3,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[4,4]]
 => [[4,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1],[4]]
 => [[1,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2],[4]]
 => [[2,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3],[4]]
 => [[3,4]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,3]]
 => [[1,1,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1,2,3]]
 => [[1,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1,3,3]]
 => [[1,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[2,2,3]]
 => [[2,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[2,3,3]]
 => [[2,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[3,3,3]]
 => [[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1,1],[3]]
 => [[1,1,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,3],[3]]
 => [[1,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[2,2],[3]]
 => [[2,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[2,3],[3]]
 => [[2,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,3} + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,1,1,2]]
 => [[1,1,1,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,1,2,2]]
 => [[1,1,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,2,2,2]]
 => [[1,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[2,2,2,2]]
 => [[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,1,1],[2]]
 => [[1,1,1,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,1,2],[2]]
 => [[1,1,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,2,2],[2]]
 => [[1,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,1],[2,2]]
 => [[1,1,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,3,4,4} + 1
[[1,5]]
 => [[1,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2,5]]
 => [[2,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3,5]]
 => [[3,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[4,5]]
 => [[4,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[5,5]]
 => [[5,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1],[5]]
 => [[1,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2],[5]]
 => [[2,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3],[5]]
 => [[3,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[4],[5]]
 => [[4,5]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,4]]
 => [[1,1,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,2,4]]
 => [[1,2,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,3,4]]
 => [[1,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,4,4]]
 => [[1,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,2,4]]
 => [[2,2,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,3,4]]
 => [[2,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,4,4]]
 => [[2,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[3,3,4]]
 => [[3,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[3,4,4]]
 => [[3,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[4,4,4]]
 => [[4,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,1],[4]]
 => [[1,1,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,2],[4]]
 => [[1,2,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,4],[2]]
 => [[1,2],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,3],[4]]
 => [[1,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1,4],[3]]
 => [[1,3],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,4],[4]]
 => [[1,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,2],[4]]
 => [[2,2,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,3],[4]]
 => [[2,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[2,4],[3]]
 => [[2,3],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,4],[4]]
 => [[2,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[3,3],[4]]
 => [[3,3,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[3,4],[4]]
 => [[3,4,4]]
 => [3]
 => [1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,3} + 1
[[1],[2],[4]]
 => [[1,2],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1],[3],[4]]
 => [[1,3],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[2],[3],[4]]
 => [[2,3],[4]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,1,1,3]]
 => [[1,1,1,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,1,2,3]]
 => [[1,1,2,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,1,3,3]]
 => [[1,1,3,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,2,2,3]]
 => [[1,2,2,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,2,3,3]]
 => [[1,2,3,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,3,3,3]]
 => [[1,3,3,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[2,2,2,3]]
 => [[2,2,2,3]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[[1,6]]
 => [[1,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2,6]]
 => [[2,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3,6]]
 => [[3,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[4,6]]
 => [[4,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[5,6]]
 => [[5,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[6,6]]
 => [[6,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1],[6]]
 => [[1,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[2],[6]]
 => [[2,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[3],[6]]
 => [[3,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[4],[6]]
 => [[4,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[5],[6]]
 => [[5,6]]
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,5],[2]]
 => [[1,2],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,5],[3]]
 => [[1,3],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1,5],[4]]
 => [[1,4],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,5],[3]]
 => [[2,3],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[2,5],[4]]
 => [[2,4],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[3,5],[4]]
 => [[3,4],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[[1],[2],[5]]
 => [[1,2],[5]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000455
(load all 4 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,0}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,0}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,0}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,3,3],[3]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1],[3],[5]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1],[4],[5]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0
[[1,1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,3],[3,4]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 0
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,3,3],[3]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,2],[3,3]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1],[3],[5]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1],[4],[5]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,5],[2],[4]]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 0
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0
[[1,1,1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000379
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0}
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0}
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0}
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0}
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0}
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0}
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0}
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0}
[[4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0}
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,3,4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,4,4]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[2,2,4]]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[2,3,4]]
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1],[3],[5]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1],[4],[5]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2,3],[2],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,3,3],[2],[3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1],[3],[5]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1],[4],[5]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2],[2],[5]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,4],[2,3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[[1,1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[[1,1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[[1,2,2],[2],[4]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => 0
Description
The number of Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$. Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000478
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[2,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0}
[[1,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,1,1],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
Description
Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000512
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[2,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0}
[[1,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,1,1],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000566
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[2,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0}
[[1,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[2,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,1,1],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,1,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000567
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[2,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0}
[[1,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[[1,1,1],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
Description
The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000620
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[2,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0}
[[1,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[2,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[3,3]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0}
[[1],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[2],[3]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0}
[[1,1,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[2,2,2]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,3}
[[1,1],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,2],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,3}
[[1,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[2,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[3,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[4,4]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0}
[[1],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[2],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[3],[4]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0}
[[1,1,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[3,3,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[2,3],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,3,3,3}
[[1,1,1,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[2,2,2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,1],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,2,2],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,3,4,4}
[[1,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[5,5]]
 => [2]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[2],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[3],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[4],[5]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,4],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,5],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,1,1],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,2],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,3],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000101The cocharge of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.