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Identifier

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001934: Integer partitions ⟶ ℤ

Values

([],1) => [1] => 1

([],2) => [1,1] => 1

([(0,1)],2) => [2] => 1

([],3) => [1,1,1] => 1

([(1,2)],3) => [2,1] => 1

([(0,1),(0,2)],3) => [2,1] => 1

([(0,2),(2,1)],3) => [3] => 2

([(0,2),(1,2)],3) => [2,1] => 1

([],4) => [1,1,1,1] => 1

([(2,3)],4) => [2,1,1] => 1

([(1,2),(1,3)],4) => [2,1,1] => 1

([(0,1),(0,2),(0,3)],4) => [2,1,1] => 1

([(0,2),(0,3),(3,1)],4) => [3,1] => 2

([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => 2

([(1,2),(2,3)],4) => [3,1] => 2

([(0,3),(3,1),(3,2)],4) => [3,1] => 2

([(1,3),(2,3)],4) => [2,1,1] => 1

([(0,3),(1,3),(3,2)],4) => [3,1] => 2

([(0,3),(1,3),(2,3)],4) => [2,1,1] => 1

([(0,3),(1,2)],4) => [2,2] => 1

([(0,3),(1,2),(1,3)],4) => [2,2] => 1

([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => 1

([(0,3),(2,1),(3,2)],4) => [4] => 5

([(0,3),(1,2),(2,3)],4) => [3,1] => 2

([],5) => [1,1,1,1,1] => 1

([(3,4)],5) => [2,1,1,1] => 1

([(2,3),(2,4)],5) => [2,1,1,1] => 1

([(1,2),(1,3),(1,4)],5) => [2,1,1,1] => 1

([(0,1),(0,2),(0,3),(0,4)],5) => [2,1,1,1] => 1

([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1] => 2

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,1,1] => 2

([(1,3),(1,4),(4,2)],5) => [3,1,1] => 2

([(0,3),(0,4),(4,1),(4,2)],5) => [3,1,1] => 2

([(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [4,1] => 5

([(0,3),(0,4),(3,2),(4,1)],5) => [3,2] => 2

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => 2

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => 2

([(2,3),(3,4)],5) => [3,1,1] => 2

([(1,4),(4,2),(4,3)],5) => [3,1,1] => 2

([(0,4),(4,1),(4,2),(4,3)],5) => [3,1,1] => 2

([(2,4),(3,4)],5) => [2,1,1,1] => 1

([(1,4),(2,4),(4,3)],5) => [3,1,1] => 2

([(0,4),(1,4),(4,2),(4,3)],5) => [3,2] => 2

([(1,4),(2,4),(3,4)],5) => [2,1,1,1] => 1

([(0,4),(1,4),(2,4),(4,3)],5) => [3,1,1] => 2

([(0,4),(1,4),(2,4),(3,4)],5) => [2,1,1,1] => 1

([(0,4),(1,4),(2,3)],5) => [2,2,1] => 1

([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => 5

([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => 2

([(0,4),(1,4),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => 2

([(1,4),(2,3)],5) => [2,2,1] => 1

([(1,4),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,4),(1,2),(1,4),(2,3)],5) => [3,2] => 2

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => 2

([(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => 1

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [3,2] => 2

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [3,2] => 2

([(0,4),(1,2),(1,4),(4,3)],5) => [3,2] => 2

([(0,4),(1,2),(1,3)],5) => [2,2,1] => 1

([(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => 1

([(0,2),(0,4),(3,1),(4,3)],5) => [4,1] => 5

([(0,4),(1,2),(1,3),(3,4)],5) => [3,1,1] => 2

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [4,1] => 5

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,3),(0,4),(1,2),(1,4)],5) => [2,2,1] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => 1

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [3,2] => 2

([(0,3),(1,2),(1,4),(3,4)],5) => [3,2] => 2

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => 2

([(1,4),(3,2),(4,3)],5) => [4,1] => 5

([(0,3),(3,4),(4,1),(4,2)],5) => [4,1] => 5

([(1,4),(2,3),(3,4)],5) => [3,1,1] => 2

([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => 5

([(0,3),(1,4),(4,2)],5) => [3,2] => 2

([(0,4),(3,2),(4,1),(4,3)],5) => [4,1] => 5

([(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => 2

([(0,4),(2,3),(3,1),(4,2)],5) => [5] => 14

([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => 2

([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => 5

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [4,1] => 5

([],6) => [1,1,1,1,1,1] => 1

([(4,5)],6) => [2,1,1,1,1] => 1

([(3,4),(3,5)],6) => [2,1,1,1,1] => 1

([(2,3),(2,4),(2,5)],6) => [2,1,1,1,1] => 1

([(1,2),(1,3),(1,4),(1,5)],6) => [2,1,1,1,1] => 1

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => [2,1,1,1,1] => 1

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => [3,1,1,1] => 2

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => [3,1,1,1] => 2

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => 2

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => 2

([(1,3),(1,4),(1,5),(5,2)],6) => [3,1,1,1] => 2

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => [3,1,1,1] => 2

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => [3,1,1,1] => 2

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => 2

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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

Greene-Kleitman invariant

Description

The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.