Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000173
Mapping
St000173: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 1
[[2,2]]
 => 1
[[1],[2]]
 => 0
[[1,3]]
 => 1
[[2,3]]
 => 2
[[3,3]]
 => 1
[[1],[3]]
 => 1
[[2],[3]]
 => 2
[[1,1,2]]
 => 1
[[1,2,2]]
 => 1
[[2,2,2]]
 => 1
[[1,1],[2]]
 => 0
[[1,2],[2]]
 => 1
[[1,4]]
 => 1
[[2,4]]
 => 2
[[3,4]]
 => 2
[[4,4]]
 => 1
[[1],[4]]
 => 1
[[2],[4]]
 => 2
[[3],[4]]
 => 2
[[1,1,3]]
 => 1
[[1,2,3]]
 => 2
[[1,3,3]]
 => 1
[[2,2,3]]
 => 2
[[2,3,3]]
 => 2
[[3,3,3]]
 => 1
[[1,1],[3]]
 => 1
[[1,2],[3]]
 => 2
[[1,3],[2]]
 => 1
[[1,3],[3]]
 => 2
[[2,2],[3]]
 => 2
[[2,3],[3]]
 => 3
[[1],[2],[3]]
 => 0
[[1,1,1,2]]
 => 1
[[1,1,2,2]]
 => 1
[[1,2,2,2]]
 => 1
[[2,2,2,2]]
 => 1
[[1,1,1],[2]]
 => 0
[[1,1,2],[2]]
 => 1
[[1,2,2],[2]]
 => 1
[[1,1],[2,2]]
 => 0
[[1,1,4]]
 => 1
[[1,2,4]]
 => 2
[[1,3,4]]
 => 2
[[1,4,4]]
 => 1
[[2,2,4]]
 => 2
[[2,3,4]]
 => 3
[[2,4,4]]
 => 2
[[3,3,4]]
 => 2
[[3,4,4]]
 => 2
Description
The segment statistic of a semistandard tableau. Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[[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)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 2
[[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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(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
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2
[[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)
 => ? = 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[[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)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 2
[[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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[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)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1
[[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)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[[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)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 2
[[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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ? = 1
[[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)
 => ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
 => ? = 2
[[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)
 => ([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(0,12),(1,10),(1,11),(2,11),(2,14),(3,10),(3,13),(4,8),(4,15),(5,9),(5,16),(6,15),(6,16),(6,17),(7,13),(7,14),(7,18),(8,12),(8,18),(9,12),(9,18),(10,17),(11,17),(13,15),(13,17),(14,16),(14,17),(15,18),(16,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(0,13),(1,12),(2,9),(2,15),(3,8),(3,14),(4,10),(4,16),(5,11),(5,17),(6,16),(6,17),(6,18),(7,14),(7,15),(7,19),(8,12),(8,18),(9,12),(9,18),(10,13),(10,19),(11,13),(11,19),(14,16),(14,18),(15,17),(15,18),(16,19),(17,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[[1,3],[4]]
 => ([(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
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 2
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 3
[[2,4],[3]]
 => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 3
[[2,4],[4]]
 => ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
 => ([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
 => ? = 3
[[3,3],[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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ? = 2
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1],[2,3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1],[2],[4]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2,3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,3],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[4]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,1],[2,2],[4]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2],[3],[4]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1,1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1],[2,3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2],[2],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2,2,3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
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: St001232
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ 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: 29%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[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)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[[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)
 => [7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[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)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[[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)
 => [7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 2
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[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)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[[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)
 => [7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[[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)
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 2
[[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)
 => [7,4,3]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => [9,6,4]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => [10,6,4]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[[2,2],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[[2,3],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[[2,4],[3]]
 => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
 => [6,4,1]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[[2,4],[4]]
 => ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
 => [7,5,1]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[[3,3],[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)
 => [7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[[1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2,3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,2],[2],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1],[2],[3],[4]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2,3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,2],[2],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2,2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1],[2,3],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,2],[2,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2,2],[4]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2],[3],[4]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,2],[2],[3],[4]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1],[2,3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1,1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,2],[2],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2,2,3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,1],[2,2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
[[1,1],[2,2],[3,3]]
 => ([],1)
 => [1]
 => [1,0]
 => 0
[[1,1,1,1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[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)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[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)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000302
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[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)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[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)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
Description
The Gutman (or modified Schultz) index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2
[[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)
 => ? = 3
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 2
[[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)
 => ? = 1
[[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)
 => ? = 2
[[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)
 => ? = 3
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 0
[[1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 0
Description
The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
Matching statistic: St000771
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 + 1
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 1 + 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2 + 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 1 + 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 + 1
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 + 1
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 + 1
[[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)
 => ? = 3 + 1
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 + 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 3 + 1
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2 + 1
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2 + 1
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 + 1
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1 + 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2 + 1
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 3 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 + 1
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 1 + 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2 + 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 1 + 1
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 + 1
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 + 1
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 + 1
[[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)
 => ? = 3 + 1
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 + 1
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 + 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 1 + 1
[[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)
 => ? = 2 + 1
[[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)
 => ? = 3 + 1
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2 + 1
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 2 + 1
[[3,4,4]]
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 + 1
[[4,4,4]]
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1 + 1
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 + 1
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 + 1
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 + 1
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 2 + 1
[[1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2,2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3,3]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1],[2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1],[2,2],[3],[4]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[1,1,1,1,1]]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.