Your data matches 107 different statistics following compositions of up to 3 maps.
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Matching statistic: St001631
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001631: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1],[2]]
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[[1],[3]]
 => ([(0,1)],2)
 => 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1,2]]
 => ([(0,1)],2)
 => 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[1,1],[2]]
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[[1,1,1],[2]]
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1],[2,2]]
 => ([],1)
 => 0
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[[1,1,1,1],[2]]
 => ([],1)
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[1,1,1],[2,2]]
 => ([],1)
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[[1,1,1,1,1],[2]]
 => ([],1)
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[[1,1,1,1],[2,2]]
 => ([],1)
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => 0
Description
The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
Matching statistic: St001479
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 5
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 5
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 6
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
Description
The number of bridges of a graph. A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St000835
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 3 = 1 + 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => 4 = 2 + 2
[[1],[3]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => 8 = 6 + 2
[[1,1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
Description
The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St000992
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 3 = 1 + 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => 4 = 2 + 2
[[1],[3]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => 8 = 6 + 2
[[1,1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St001055
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 3 = 1 + 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => 4 = 2 + 2
[[1],[3]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7 = 5 + 2
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => 8 = 6 + 2
[[1,1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 3 + 2
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6 = 4 + 2
[[1,1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 3 = 1 + 2
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4 = 2 + 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => [2]
 => 2 = 0 + 2
Description
The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
Matching statistic: St000714
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1],[2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 4 = 1 + 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => 5 = 2 + 3
[[1],[3]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[1,1],[2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 7 = 4 + 3
[[1,1,1],[2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1],[2,2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 7 = 4 + 3
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 8 = 5 + 3
[[1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 7 = 4 + 3
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 8 = 5 + 3
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => 9 = 6 + 3
[[1,1,1,1,1],[2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 6 = 3 + 3
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 7 = 4 + 3
[[1,1,1,1],[2,2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [3]
 => 4 = 1 + 3
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 5 = 2 + 3
[[1,1,1],[2,2,2]]
 => ([],1)
 => [2]
 => 3 = 0 + 3
Description
The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St001189
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 98%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 5
[[1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 5
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[[1,1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[1,1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 0
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2}
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2}
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001330
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2} + 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2} + 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001014
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001014: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 94%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[[2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5 + 1
[[1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1,1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[1,1,2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,2,2,2,2,2]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {5,6} + 1
[[2,2,2,2,2,2]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {5,6} + 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1,2,2,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[[1,2,2,2,2],[2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,1,2,2],[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
Description
Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000080The rank of the poset. St000189The number of elements in the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001621The number of atoms of a lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000456The monochromatic index 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. St000464The Schultz index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001645The pebbling number of a connected graph. 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 $eA$. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000947The major index east count of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St000997The even-odd crank of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001808The box weight or horizontal decoration of a Dyck path.