- FindStat

Identifier

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤ

Values

[1] => [1] => 0

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

[1,1] => [2] => 1

[3] => [3] => 0

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

[1,1,1] => [2,1] => 1

[4] => [1,1,1,1] => 0

[3,1] => [3,1] => 0

[2,2] => [4] => 2

[2,1,1] => [2,1,1] => 1

[1,1,1,1] => [2,2] => 1

[5] => [5] => 0

[4,1] => [1,1,1,1,1] => 0

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

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

[2,2,1] => [4,1] => 2

[2,1,1,1] => [2,1,1,1] => 1

[1,1,1,1,1] => [2,2,1] => 1

[6] => [3,3] => 0

[5,1] => [5,1] => 0

[4,2] => [1,1,1,1,1,1] => 0

[4,1,1] => [2,1,1,1,1] => 1

[3,3] => [6] => 3

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

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

[2,2,2] => [4,1,1] => 2

[2,2,1,1] => [4,2] => 2

[2,1,1,1,1] => [2,2,1,1] => 1

[1,1,1,1,1,1] => [2,2,2] => 1

[7] => [7] => 0

[6,1] => [3,3,1] => 0

[5,2] => [5,1,1] => 0

[5,1,1] => [5,2] => 1

[4,3] => [3,1,1,1,1] => 0

[4,2,1] => [1,1,1,1,1,1,1] => 0

[4,1,1,1] => [2,1,1,1,1,1] => 1

[3,3,1] => [6,1] => 3

[3,2,2] => [4,3] => 2

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

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

[2,2,2,1] => [4,1,1,1] => 2

[2,2,1,1,1] => [4,2,1] => 2

[2,1,1,1,1,1] => [2,2,1,1,1] => 1

[1,1,1,1,1,1,1] => [2,2,2,1] => 1

[8] => [1,1,1,1,1,1,1,1] => 0

[7,1] => [7,1] => 0

[6,2] => [3,3,1,1] => 0

[6,1,1] => [3,3,2] => 1

[5,3] => [5,3] => 0

[5,2,1] => [5,1,1,1] => 0

[5,1,1,1] => [5,2,1] => 1

[4,4] => [8] => 4

[4,3,1] => [3,1,1,1,1,1] => 0

[4,2,2] => [4,1,1,1,1] => 2

[4,2,1,1] => [2,1,1,1,1,1,1] => 1

[4,1,1,1,1] => [2,2,1,1,1,1] => 1

[3,3,2] => [6,1,1] => 3

[3,3,1,1] => [6,2] => 3

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

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

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

[2,2,2,2] => [4,4] => 2

[2,2,2,1,1] => [4,2,1,1] => 2

[2,2,1,1,1,1] => [4,2,2] => 2

[2,1,1,1,1,1,1] => [2,2,2,1,1] => 1

[1,1,1,1,1,1,1,1] => [2,2,2,2] => 1

[9] => [9] => 0

[8,1] => [1,1,1,1,1,1,1,1,1] => 0

[7,2] => [7,1,1] => 0

[7,1,1] => [7,2] => 1

[6,3] => [3,3,3] => 0

[6,2,1] => [3,3,1,1,1] => 0

[6,1,1,1] => [3,3,2,1] => 1

[5,4] => [5,1,1,1,1] => 0

[5,3,1] => [5,3,1] => 0

[5,2,2] => [5,4] => 2

[5,2,1,1] => [5,2,1,1] => 1

[5,1,1,1,1] => [5,2,2] => 1

[4,4,1] => [8,1] => 4

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

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

[4,2,2,1] => [4,1,1,1,1,1] => 2

[4,2,1,1,1] => [2,1,1,1,1,1,1,1] => 1

[4,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1

[3,3,3] => [6,3] => 3

[3,3,2,1] => [6,1,1,1] => 3

[3,3,1,1,1] => [6,2,1] => 3

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

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

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

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

[2,2,2,2,1] => [4,4,1] => 2

[2,2,2,1,1,1] => [4,2,1,1,1] => 2

[2,2,1,1,1,1,1] => [4,2,2,1] => 2

[2,1,1,1,1,1,1,1] => [2,2,2,1,1,1] => 1

[1,1,1,1,1,1,1,1,1] => [2,2,2,2,1] => 1

[10] => [5,5] => 0

[9,1] => [9,1] => 0

[8,2] => [1,1,1,1,1,1,1,1,1,1] => 0

[8,1,1] => [2,1,1,1,1,1,1,1,1] => 1

[7,3] => [7,3] => 0

>>> Load all 1200 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q$

$F_{4} = 2 + 2\ q + q^{2}$

$F_{5} = 3 + 3\ q + q^{2}$

$F_{6} = 4 + 4\ q + 2\ q^{2} + q^{3}$

$F_{7} = 5 + 6\ q + 3\ q^{2} + q^{3}$

$F_{8} = 6 + 8\ q + 5\ q^{2} + 2\ q^{3} + q^{4}$

$F_{9} = 8 + 11\ q + 7\ q^{2} + 3\ q^{3} + q^{4}$

$F_{10} = 10 + 14\ q + 10\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$

$F_{11} = 12 + 19\ q + 14\ q^{2} + 7\ q^{3} + 3\ q^{4} + q^{5}$

$F_{12} = 15 + 24\ q + 19\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$

$F_{13} = 18 + 31\ q + 26\ q^{2} + 15\ q^{3} + 7\ q^{4} + 3\ q^{5} + q^{6}$

$F_{14} = 22 + 39\ q + 34\ q^{2} + 21\ q^{3} + 11\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{7}$

$F_{15} = 27 + 49\ q + 45\ q^{2} + 29\ q^{3} + 15\ q^{4} + 7\ q^{5} + 3\ q^{6} + q^{7}$

$F_{16} = 32 + 61\ q + 58\ q^{2} + 39\ q^{3} + 22\ q^{4} + 11\ q^{5} + 5\ q^{6} + 2\ q^{7} + q^{8}$

Description

Half of the largest even part of an integer partition.
The largest even part is recorded by St000995The largest even part of an integer partition..

Map

Glaisher-Franklin inverse

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].