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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St001615
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(load all 3 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(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
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(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
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(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: St000080
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
Description
The rank of the poset.
Matching statistic: St001637
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000528
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000680
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The Grundy value for Hackendot on posets.
Two players take turns and remove an order filter. The player who is faced with the one element poset looses. This game is a slight variation of Chomp.
This statistic is the Grundy value of the poset, that is, the smallest non-negative integer which does not occur as value of a poset obtained by a single move.
Matching statistic: St000717
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The number of ordinal summands of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
Matching statistic: St000906
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The length of the shortest maximal chain in a poset.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000912The number of maximal antichains in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001875The number of simple modules with projective dimension at most 1. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000656The number of cuts of a poset.
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