Identifier
-
Mp00065:
Permutations
—permutation poset⟶
Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001934: Integer partitions ⟶ ℤ
Values
[1] => ([],1) => [1] => [1] => 1
[1,2] => ([(0,1)],2) => [1] => [1] => 1
[2,1] => ([],2) => [2] => [1,1] => 1
[1,2,3] => ([(0,2),(2,1)],3) => [1] => [1] => 1
[1,3,2] => ([(0,1),(0,2)],3) => [2] => [1,1] => 1
[2,1,3] => ([(0,2),(1,2)],3) => [2] => [1,1] => 1
[2,3,1] => ([(1,2)],3) => [3] => [1,1,1] => 1
[3,1,2] => ([(1,2)],3) => [3] => [1,1,1] => 1
[3,2,1] => ([],3) => [3,3] => [2,2,2] => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [1] => [1] => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => [2] => [1,1] => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => [2] => [1,1] => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => [3] => [1,1,1] => 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => [3] => [1,1,1] => 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => [3,3] => [2,2,2] => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => [2] => [1,1] => 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [2,2] => 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1] => 1
[2,3,4,1] => ([(1,2),(2,3)],4) => [4] => [1,1,1,1] => 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => [3,2] => [2,2,1] => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1] => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => [3,2] => [2,2,1] => 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => [3,3] => [2,2,2] => 1
[3,4,1,2] => ([(0,3),(1,2)],4) => [4,2] => [2,2,1,1] => 1
[4,1,2,3] => ([(1,2),(2,3)],4) => [4] => [1,1,1,1] => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => [1] => [1] => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => [2] => [1,1] => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [2] => [1,1] => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => [3] => [1,1,1] => 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => [3] => [1,1,1] => 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => [3,3] => [2,2,2] => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [2] => [1,1] => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [2,2] => [2,2] => 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => [1,1,1] => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => [1,1,1,1] => 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [2,2,1] => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => [1,1,1] => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [2,2,1] => 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,3] => [2,2,2] => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => [2,2,1,1] => 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => [1,1,1,1] => 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => [2] => [1,1] => 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => [2,2] => [2,2] => 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [2,2] => [2,2] => 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [6] => [1,1,1,1,1,1] => 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [6] => [1,1,1,1,1,1] => 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => [1,1,1] => 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1] => 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => [1,1,1,1] => 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => [5] => [1,1,1,1,1] => 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => [7] => [1,1,1,1,1,1,1] => 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [2,2,1] => 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => [7] => [1,1,1,1,1,1,1] => 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => [1,1,1] => 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1] => 1
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [2,2,1] => 1
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => [7] => [1,1,1,1,1,1,1] => 1
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => [3,3] => [2,2,2] => 1
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => [2,2,1,1] => 1
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => [1,1,1,1] => 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => [7] => [1,1,1,1,1,1,1] => 1
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => [5] => [1,1,1,1,1] => 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => [1] => 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [2] => [1,1] => 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [2] => [1,1] => 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [3] => [1,1,1] => 1
[1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [3] => [1,1,1] => 1
[1,2,3,6,5,4] => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [3,3] => [2,2,2] => 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [2] => [1,1] => 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6) => [2,2] => [2,2] => 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [3] => [1,1,1] => 1
[1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => [1,1,1,1] => 1
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => [2,2,1] => 1
[1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [3] => [1,1,1] => 1
[1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => [2,2,1] => 1
[1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [3,3] => [2,2,2] => 1
[1,2,5,6,3,4] => ([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [4,2] => [2,2,1,1] => 1
[1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => [1,1,1,1] => 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [2] => [1,1] => 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [2,2] => [2,2] => 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [2,2] => [2,2] => 1
[1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [6] => [1,1,1,1,1,1] => 1
[1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [6] => [1,1,1,1,1,1] => 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [3] => [1,1,1] => 1
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [6] => [1,1,1,1,1,1] => 1
[1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [4] => [1,1,1,1] => 1
[1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => [1,1,1,1,1] => 1
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => [7] => [1,1,1,1,1,1,1] => 1
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => [2,2,1] => 1
[1,3,6,2,4,5] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => [7] => [1,1,1,1,1,1,1] => 1
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [3] => [1,1,1] => 1
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [6] => [1,1,1,1,1,1] => 1
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => [2,2,1] => 1
[1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => [7] => [1,1,1,1,1,1,1] => 1
[1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [3,3] => [2,2,2] => 1
[1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [4,2] => [2,2,1,1] => 1
[1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [4] => [1,1,1,1] => 1
[1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => [7] => [1,1,1,1,1,1,1] => 1
[1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => [1,1,1,1,1] => 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [2] => [1,1] => 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [2,2] => [2,2] => 1
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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 $(1,2)(2,3)$.
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
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a poset.
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