searching the database
Your data matches 335 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001333
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St001354
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
Description
The number of series nodes in the modular decomposition of a graph.
Matching statistic: St001393
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> ([],5)
=> ([],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000208
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([],3)
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [2,2]
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1]
=> [1]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> [2]
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [2,2,1]
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [2,2]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [2,2]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [2,2]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 2 = 1 + 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000299
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The number of nonisomorphic vertex-induced subtrees.
Matching statistic: St001093
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Matching statistic: St001674
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The number of vertices of the largest induced star graph in the graph.
Matching statistic: St000025
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> 0
([],2)
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1,0]
=> 1
([],3)
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([],4)
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([],5)
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2}
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000676
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> 0
([],2)
=> []
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1]
=> [1,0]
=> 1
([],3)
=> []
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([],4)
=> []
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([],5)
=> []
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2}
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000481
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? = 1
([(0,1)],2)
=> [1]
=> [1]
=> 0
([],3)
=> []
=> ?
=> ? = 1
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [2,1]
=> 1
([],4)
=> []
=> ?
=> ? = 2
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,1]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [2,1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [2,2]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [2,2,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [2,2,2]
=> 1
([],5)
=> []
=> ?
=> ? = 2
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [2,2]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [3,2]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [2,2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,2,1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [2,2,2]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [2,2,2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3,2,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [2,2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [2,2,2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [2,2,2,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [2,2,2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [3,2,2]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [2,2,2,2]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [2,2,2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [2,2,2,2]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [2,2,2,2,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [2,2,2,2,2]
=> 1
Description
The number of upper covers of a partition in dominance order.
The following 325 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001271The competition number of a graph. St000706The product of the factorials of the multiplicities of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001363The Euler characteristic of a graph according to Knill. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000383The last part of an integer composition. St000753The Grundy value for the game of Kayles on a binary word. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000775The multiplicity of the largest eigenvalue in a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000306The bounce count of a Dyck path. St000026The position of the first return of a Dyck path. St000287The number of connected components of a graph. St000544The cop number of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000916The packing number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000785The number of distinct colouring schemes of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001471The magnitude of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000260The radius of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001732The number of peaks visible from the left. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000015The number of peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000618The number of self-evacuating tableaux of given shape. St000658The number of rises of length 2 of a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St000874The position of the last double rise in a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001031The height of 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001280The number of parts of an integer partition that are at least two. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001530The depth of a Dyck path. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001808The box weight or horizontal decoration of a Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001955The number of natural descents for set-valued two row standard Young tableaux. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001261The Castelnuovo-Mumford regularity of a graph. St000284The Plancherel distribution on integer partitions. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000762The sum of the positions of the weak records of an integer composition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St000553The number of blocks of a graph. St000741The Colin de Verdière graph invariant. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000120The number of left tunnels of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001282The number of graphs with the same chromatic polynomial. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001571The Cartan determinant of the integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000454The largest eigenvalue of a graph if it is integral. St000928The sum of the coefficients of the character polynomial of an integer partition. St000456The monochromatic index of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001281The normalized isoperimetric number of a graph. St001389The number of partitions of the same length below the given integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000088The row sums of the character table of the symmetric group. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. St001871The number of triconnected components of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001734The lettericity of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001520The number of strict 3-descents. St001877Number of indecomposable injective modules with projective dimension 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000914The sum of the values of the Möbius function of a poset. 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. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000735The last entry on the main diagonal of a standard tableau. 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. St000422The energy of a graph, if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001651The Frankl number of a lattice. 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. St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!