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Your data matches 240 different statistics following compositions of up to 3 maps.
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Matching statistic: St000268
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
Description
The number of strongly connected orientations of a graph.
Matching statistic: St000344
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St001073
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
Description
The number of nowhere zero 3-flows of a graph.
This is the absolute value of the evaluation of the Tutte polynomial of the graph at $(x,y)=(0,-2)$.
For $4$-regular graphs, this coincides with the number of Eulerian orientations.
Matching statistic: St001477
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
Description
The number of nowhere zero 5-flows of a graph.
Matching statistic: St001478
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
Description
The number of nowhere zero 4-flows of a graph.
Matching statistic: St001675
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => 1
([],2)
=> ([],2)
=> [2] => 1
([(0,1)],2)
=> ([],1)
=> [1] => 1
([],3)
=> ([],3)
=> [3] => 1
([(1,2)],3)
=> ([],2)
=> [2] => 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1] => 1
([],4)
=> ([],4)
=> [4] => 1
([(2,3)],4)
=> ([],3)
=> [3] => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,2)],4)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1] => 1
([],5)
=> ([],5)
=> [5] => 1
([(3,4)],5)
=> ([],4)
=> [4] => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,3)],5)
=> ([],3)
=> [3] => 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [3] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [4,3] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1] => 1
Description
The number of parts equal to the part in the reversed composition.
Matching statistic: St001795
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(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)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(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)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
Description
The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).
Matching statistic: St001442
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00250: Graphs —clique graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001442: 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,1)],2)
=> ([],1)
=> [1]
=> [1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> [1]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
Matching statistic: St001593
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001593: 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,1)],2)
=> ([],1)
=> [1]
=> [1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> [1]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 1
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St000309
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(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)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(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)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
Description
The number of vertices with even degree.
The following 230 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000315The number of isolated vertices of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001691The number of kings in a graph. St001972The a-number of a graph. St000302The determinant of the distance matrix of a connected graph. St000351The determinant of the adjacency matrix of a graph. St000096The number of spanning trees of a graph. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000450The number of edges minus the number of vertices plus 2 of a graph. St000948The chromatic discriminant of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000137The Grundy value of an integer partition. St000390The number of runs of ones in a binary word. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001498The normalised height of a Nakayama algebra with magnitude 1. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000928The sum of the coefficients of the character polynomial of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000379The number of Hamiltonian cycles in a graph. St000456The monochromatic index of a connected graph. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. 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. 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. St000389The number of runs of ones of odd length in a binary word. St000478Another weight of a partition according to Alladi. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000628The balance of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. 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. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000934The 2-degree of an integer partition. St000935The number of ordered refinements of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. 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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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. 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. 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 $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001651The Frankl number of a lattice. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000914The sum of the values of the Möbius function of a poset. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000735The last entry on the main diagonal of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001541The Gini index of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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.
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