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Your data matches 415 different statistics following compositions of up to 3 maps.
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Matching statistic: St001615
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Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
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(load all 3 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$
v(a\vee b) + v(a\wedge b) = v(a) + v(b).
$$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]].
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001622
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 2
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001719
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001875
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001845
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St000080
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
Description
The rank of the poset.
Matching statistic: St000189
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
Description
The number of elements in the poset.
Matching statistic: St001613
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001618
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(2,7),(3,7),(4,7),(5,7),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,7),(5,1),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,7),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(4,7),(5,7),(6,1),(6,2),(7,3)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,7),(3,5),(4,5),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(2,7),(3,7),(4,6),(5,6),(6,7),(7,1)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,7),(4,3),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,1),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,6),(5,7),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The cardinality of the Frattini sublattice of a lattice.
The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.
The following 405 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001881The number of factors of a lattice as a Cartesian product of lattices. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000528The height of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000908The length of the shortest maximal antichain in a poset. St000912The number of maximal antichains in a poset. St000914The sum of the values of the Möbius function of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001625The Möbius invariant of a lattice. St001664The number of non-isomorphic subposets of a poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001718The number of non-empty open intervals in a poset. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001890The maximum magnitude of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000643The size of the largest orbit of antichains under Panyushev complementation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001616The number of neutral elements in a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St001754The number of tolerances of a finite lattice. St001782The order of rowmotion on the set of order ideals of a poset. St001846The number of elements which do not have a complement in the lattice. St001909The number of interval-closed sets of a poset. St001964The interval resolution global dimension of a poset. St001619The number of non-isomorphic sublattices of a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000100The number of linear extensions of a poset. St000171The degree of the graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St000259The diameter of a connected graph. St000271The chromatic index of a graph. St000286The number of connected components of the complement of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000636The hull number of a graph. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000644The number of graphs with given frequency partition. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001029The size of the core of a graph. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001268The size of the largest ordinal summand in the poset. St001286The annihilation number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001399The distinguishing number of a poset. St001458The rank of the adjacency matrix of a graph. St001494The Alon-Tarsi number of a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001512The minimum rank of a graph. St001581The achromatic number of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001642The Prague dimension of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001703The villainy of a graph. St001716The 1-improper chromatic number of a graph. St001734The lettericity of a graph. St001765The number of connected components of the friends and strangers graph. St001779The order of promotion on the set of linear extensions of a poset. St001902The number of potential covers of a poset. St001917The order of toric promotion on the set of labellings of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001949The rigidity index of a graph. St000147The largest part of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000256The number of parts from which one can substract 2 and still get an integer partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000345The number of refinements of a partition. St000349The number of different adjacency matrices of a graph. St000384The maximal part of the shifted composition of an integer partition. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000482The (zero)-forcing number of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000632The jump number of the poset. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000708The product of the parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000784The maximum of the length and the largest part of the integer partition. St000785The number of distinct colouring schemes of a graph. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000897The number of different multiplicities of parts of an integer partition. St000911The number of maximal antichains of maximal size in a poset. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000993The multiplicity of the largest part of an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St001093The detour number of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001272The number of graphs with the same degree sequence. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001282The number of graphs with the same chromatic polynomial. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001391The disjunction number of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001463The number of distinct columns in the nullspace of a graph. St001472The permanent of the Coxeter matrix of the poset. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001674The number of vertices of the largest induced star graph in the graph. St001694The number of maximal dissociation sets in a graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001812The biclique partition number of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St001957The number of Hasse diagrams with a given underlying undirected graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. 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. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000313The number of degree 2 vertices of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000327The number of cover relations in a poset. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000403The Szeged index minus the Wiener index of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000455The second largest eigenvalue of a graph if it is integral. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000552The number of cut vertices of a graph. St000635The number of strictly order preserving maps of a poset into itself. St000637The length of the longest cycle in a graph. St000656The number of cuts of a poset. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. 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. St000926The clique-coclique number of a graph. St000929The constant term of the character polynomial of an integer partition. St000948The chromatic discriminant of a graph. St000995The largest even part of an integer partition. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001119The length of a shortest maximal path in a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001248Sum of the even parts of a partition. St001252Half the sum of the even parts of a partition. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. 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. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001367The smallest number which does not occur as degree of a vertex in a graph. St001374The Padmakar-Ivan index of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001561The value of the elementary symmetric function evaluated at 1. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001638The book thickness of a graph. St001645The pebbling number of a connected graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001717The largest size of an interval in a poset. St001736The total number of cycles in a graph. St001764The number of non-convex subsets of vertices in a graph. St001783The number of odd automorphisms of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000450The number of edges minus the number of vertices plus 2 of a graph. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000707The product of the factorials of the parts. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St001129The product of the squares of the parts of a partition. St000180The number of chains of a poset. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset. St001709The number of homomorphisms to the three element chain of a poset. St000634The number of endomorphisms of a poset. St000264The girth of a graph, which is not a tree. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000422The energy of a graph, if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000655The length of the minimal rise of a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000921The number of internal inversions of a binary word. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001593This is the number of standard Young tableaux of the given shifted shape. St000667The greatest common divisor of the parts of the partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000627The exponent of a binary word. St000296The length of the symmetric border of a binary word. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001696The natural major index of a standard Young tableau. St001256Number of simple reflexive modules that are 2-stable reflexive. St001481The minimal height of a peak of a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
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