Identifier
-
Mp00308:
Integer partitions
—Bulgarian solitaire⟶
Integer partitions
St000145: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[2] => [1,1] => -1
[1,1] => [2] => 1
[3] => [2,1] => 0
[2,1] => [2,1] => 0
[1,1,1] => [3] => 2
[4] => [3,1] => 1
[3,1] => [2,2] => 0
[2,2] => [2,1,1] => -1
[2,1,1] => [3,1] => 1
[1,1,1,1] => [4] => 3
[5] => [4,1] => 2
[4,1] => [3,2] => 1
[3,2] => [2,2,1] => -1
[3,1,1] => [3,2] => 1
[2,2,1] => [3,1,1] => 0
[2,1,1,1] => [4,1] => 2
[1,1,1,1,1] => [5] => 4
[6] => [5,1] => 3
[5,1] => [4,2] => 2
[4,2] => [3,2,1] => 0
[4,1,1] => [3,3] => 1
[3,3] => [2,2,2] => -1
[3,2,1] => [3,2,1] => 0
[3,1,1,1] => [4,2] => 2
[2,2,2] => [3,1,1,1] => -1
[2,2,1,1] => [4,1,1] => 1
[2,1,1,1,1] => [5,1] => 3
[1,1,1,1,1,1] => [6] => 5
[7] => [6,1] => 4
[6,1] => [5,2] => 3
[5,2] => [4,2,1] => 1
[5,1,1] => [4,3] => 2
[4,3] => [3,2,2] => 0
[4,2,1] => [3,3,1] => 0
[4,1,1,1] => [4,3] => 2
[3,3,1] => [3,2,2] => 0
[3,2,2] => [3,2,1,1] => -1
[3,2,1,1] => [4,2,1] => 1
[3,1,1,1,1] => [5,2] => 3
[2,2,2,1] => [4,1,1,1] => 0
[2,2,1,1,1] => [5,1,1] => 2
[2,1,1,1,1,1] => [6,1] => 4
[1,1,1,1,1,1,1] => [7] => 6
[8] => [7,1] => 5
[7,1] => [6,2] => 4
[6,2] => [5,2,1] => 2
[6,1,1] => [5,3] => 3
[5,3] => [4,2,2] => 1
[5,2,1] => [4,3,1] => 1
[5,1,1,1] => [4,4] => 2
[4,4] => [3,3,2] => 0
[4,3,1] => [3,3,2] => 0
[4,2,2] => [3,3,1,1] => -1
[4,2,1,1] => [4,3,1] => 1
[4,1,1,1,1] => [5,3] => 3
[3,3,2] => [3,2,2,1] => -1
[3,3,1,1] => [4,2,2] => 1
[3,2,2,1] => [4,2,1,1] => 0
[3,2,1,1,1] => [5,2,1] => 2
[3,1,1,1,1,1] => [6,2] => 4
[2,2,2,2] => [4,1,1,1,1] => -1
[2,2,2,1,1] => [5,1,1,1] => 1
[2,2,1,1,1,1] => [6,1,1] => 3
[2,1,1,1,1,1,1] => [7,1] => 5
[1,1,1,1,1,1,1,1] => [8] => 7
[9] => [8,1] => 6
[8,1] => [7,2] => 5
[7,2] => [6,2,1] => 3
[7,1,1] => [6,3] => 4
[6,3] => [5,2,2] => 2
[6,2,1] => [5,3,1] => 2
[6,1,1,1] => [5,4] => 3
[5,4] => [4,3,2] => 1
[5,3,1] => [4,3,2] => 1
[5,2,2] => [4,3,1,1] => 0
[5,2,1,1] => [4,4,1] => 1
[5,1,1,1,1] => [5,4] => 3
[4,4,1] => [3,3,3] => 0
[4,3,2] => [3,3,2,1] => -1
[4,3,1,1] => [4,3,2] => 1
[4,2,2,1] => [4,3,1,1] => 0
[4,2,1,1,1] => [5,3,1] => 2
[4,1,1,1,1,1] => [6,3] => 4
[3,3,3] => [3,2,2,2] => -1
[3,3,2,1] => [4,2,2,1] => 0
[3,3,1,1,1] => [5,2,2] => 2
[3,2,2,2] => [4,2,1,1,1] => -1
[3,2,2,1,1] => [5,2,1,1] => 1
[3,2,1,1,1,1] => [6,2,1] => 3
[3,1,1,1,1,1,1] => [7,2] => 5
[2,2,2,2,1] => [5,1,1,1,1] => 0
[2,2,2,1,1,1] => [6,1,1,1] => 2
[2,2,1,1,1,1,1] => [7,1,1] => 4
[2,1,1,1,1,1,1,1] => [8,1] => 6
[1,1,1,1,1,1,1,1,1] => [9] => 8
[10] => [9,1] => 7
[9,1] => [8,2] => 6
[8,2] => [7,2,1] => 4
[8,1,1] => [7,3] => 5
[7,3] => [6,2,2] => 3
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Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Map
Bulgarian solitaire
Description
A move in Bulgarian solitaire.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.
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