Identifier
Values
([],1) => ([],1) => [1] => 1
([],2) => ([],2) => [1,1] => 1
([(0,1)],2) => ([(0,1)],2) => [2] => 1
([],3) => ([],3) => [1,1,1] => 1
([(1,2)],3) => ([(1,2)],3) => [2,1] => 1
([(0,1),(0,2)],3) => ([(0,2),(1,2)],3) => [2,2] => 1
([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => [2,2] => 1
([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => [2,2] => 1
([],4) => ([],4) => [1,1,1,1] => 1
([(2,3)],4) => ([(2,3)],4) => [2,1,1] => 1
([(1,2),(1,3)],4) => ([(1,3),(2,3)],4) => [2,2,1] => 1
([(0,1),(0,2),(0,3)],4) => ([(0,3),(1,3),(2,3)],4) => [2,2,2] => 1
([(0,2),(0,3),(3,1)],4) => ([(0,3),(1,2),(2,3)],4) => [2,2,2] => 1
([(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => [2,2,1] => 1
([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => [2,2,2] => 1
([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => [2,2,1] => 1
([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => [2,2,2] => 1
([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => [2,2,2] => 1
([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => [2,2] => 1
([(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2),(2,3)],4) => [2,2,2] => 1
([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => [2,2,2] => 1
([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => [2,2,2] => 1
([],5) => ([],5) => [1,1,1,1,1] => 1
([(3,4)],5) => ([(3,4)],5) => [2,1,1,1] => 1
([(2,3),(2,4)],5) => ([(2,4),(3,4)],5) => [2,2,1,1] => 1
([(1,2),(1,3),(1,4)],5) => ([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => 1
([(1,3),(1,4),(4,2)],5) => ([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => 1
([(2,3),(3,4)],5) => ([(2,4),(3,4)],5) => [2,2,1,1] => 1
([(1,4),(4,2),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => 1
([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => [2,2,1,1] => 1
([(1,4),(2,4),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => 1
([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => 1
([(0,4),(1,4),(2,3)],5) => ([(0,1),(2,4),(3,4)],5) => [2,2,2] => 1
([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => [2,2,1] => 1
([(1,4),(2,3),(2,4)],5) => ([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => 1
([(0,4),(1,2),(1,3)],5) => ([(0,1),(2,4),(3,4)],5) => [2,2,2] => 1
([(1,4),(3,2),(4,3)],5) => ([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => 1
([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => 1
([(0,3),(1,4),(4,2)],5) => ([(0,1),(2,4),(3,4)],5) => [2,2,2] => 1
([],6) => ([],6) => [1,1,1,1,1,1] => 1
([(4,5)],6) => ([(4,5)],6) => [2,1,1,1,1] => 1
([(3,4),(3,5)],6) => ([(3,5),(4,5)],6) => [2,2,1,1,1] => 1
([(3,4),(4,5)],6) => ([(3,5),(4,5)],6) => [2,2,1,1,1] => 1
([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => [2,2,1,1,1] => 1
([(1,5),(2,5),(3,4)],6) => ([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => 1
([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => [2,2,1,1] => 1
([(1,5),(2,3),(2,4)],6) => ([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => 1
([(1,3),(2,4),(4,5)],6) => ([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => 1
([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => [2,2,2] => 1
([],7) => ([],7) => [1,1,1,1,1,1,1] => 1
([(5,6)],7) => ([(5,6)],7) => [2,1,1,1,1,1] => 1
([(3,6),(4,5)],7) => ([(3,6),(4,5)],7) => [2,2,1,1,1] => 1
([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$ (a_1, b_1),\dots,(a_r, b_r) $$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Map
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.