Identifier
-
Mp00312:
Integer partitions
—Glaisher-Franklin⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤ
Values
[1] => [1] => [1,0,1,0] => [1,2] => 0
[2] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 1
[1,1] => [2] => [1,1,0,0,1,0] => [2,1,3] => 0
[3] => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[2,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
[1,1,1] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 0
[4] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 0
[2,2] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 1
[2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[5] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
[4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 2
[3,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[3,1,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[2,2,1] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 1
[2,1,1,1] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 1
[1,1,1,1,1] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 0
[6] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 3
[5,1] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => 0
[4,2] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 2
[4,1,1] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2
[3,2,1] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 1
[3,1,1,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[2,2,1,1] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 1
[2,1,1,1,1] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 1
[1,1,1,1,1,1] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 0
[6,1] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[5,2] => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => 1
[5,1,1] => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [4,3,5,2,1,6] => 0
[4,3] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2
[4,2,1] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 2
[4,1,1,1] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
[3,2,2] => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => 1
[3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1
[3,1,1,1,1] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 0
[2,1,1,1,1,1] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
[1,1,1,1,1,1,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 4
[6,2] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 3
[6,1,1] => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 3
[5,3] => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => 0
[5,2,1] => [5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => 1
[5,1,1,1] => [5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => 0
[4,4] => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 2
[4,3,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[4,2,1,1] => [2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 2
[4,1,1,1,1] => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[3,2,1,1,1] => [3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 1
[3,1,1,1,1,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 0
[2,2,1,1,1,1] => [4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 1
[2,1,1,1,1,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1
[8,1] => [4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,2,1,6,5] => 4
[6,3] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 3
[6,2,1] => [3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 3
[6,1,1,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 3
[5,4] => [5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => 2
[5,3,1] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => 0
[5,2,2] => [5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 1
[5,2,1,1] => [5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => 1
[5,1,1,1,1] => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 0
[4,4,1] => [2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
[4,3,2] => [3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 2
[4,3,1,1] => [3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => 2
[4,1,1,1,1,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 2
[3,2,1,1,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[3,1,1,1,1,1,1] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[2,1,1,1,1,1,1,1] => [4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 1
[8,2] => [4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => 4
[8,1,1] => [4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => 4
[6,4] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => 3
[6,3,1] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => 3
[6,2,1,1] => [3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 3
[6,1,1,1,1] => [4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => 3
[5,4,1] => [5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => 2
[5,3,2] => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => 1
[5,3,1,1] => [5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => 0
[5,2,1,1,1] => [5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 1
[5,1,1,1,1,1] => [5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => 0
[4,4,2] => [2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,7,6,5,4,2] => 2
[4,3,1,1,1] => [3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 2
[4,2,1,1,1,1] => [4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 2
[4,1,1,1,1,1,1] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => 2
[3,2,1,1,1,1,1] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 1
[3,1,1,1,1,1,1,1] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[8,3] => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 4
[8,2,1] => [4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 4
[8,1,1,1] => [4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 4
[6,5] => [5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 3
[6,4,1] => [3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 3
[6,3,2] => [3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 3
[6,3,1,1] => [3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 3
[6,2,1,1,1] => [3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => 3
[6,1,1,1,1,1] => [4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 3
[5,4,2] => [5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 2
[5,4,1,1] => [5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 2
[5,3,2,1] => [5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 1
[5,3,1,1,1] => [5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 0
[5,2,1,1,1,1] => [5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => 1
[5,1,1,1,1,1,1] => [5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => 0
[4,4,1,1,1] => [2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 2
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Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 312-avoiding permutation
Description
Map
Glaisher-Franklin
Description
The Glaisher-Franklin bijection on integer partitions.
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].
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