Identifier
Values
[1] => [1] => 1
[2] => [1,1] => 1
[1,1] => [2] => 2
[3] => [3] => 1
[2,1] => [1,1,1] => 1
[1,1,1] => [2,1] => 2
[4] => [1,1,1,1] => 1
[3,1] => [3,1] => 1
[2,2] => [4] => 2
[2,1,1] => [2,1,1] => 2
[1,1,1,1] => [2,2] => 2
[5] => [5] => 1
[4,1] => [1,1,1,1,1] => 1
[3,2] => [3,1,1] => 1
[3,1,1] => [3,2] => 1
[2,2,1] => [4,1] => 2
[2,1,1,1] => [2,1,1,1] => 2
[1,1,1,1,1] => [2,2,1] => 2
[6] => [3,3] => 1
[5,1] => [5,1] => 1
[4,2] => [1,1,1,1,1,1] => 1
[4,1,1] => [2,1,1,1,1] => 2
[3,3] => [6] => 2
[3,2,1] => [3,1,1,1] => 1
[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] => 2
[1,1,1,1,1,1] => [2,2,2] => 2
[7] => [7] => 1
[6,1] => [3,3,1] => 1
[5,2] => [5,1,1] => 1
[5,1,1] => [5,2] => 1
[4,3] => [3,1,1,1,1] => 1
[4,2,1] => [1,1,1,1,1,1,1] => 1
[4,1,1,1] => [2,1,1,1,1,1] => 2
[3,3,1] => [6,1] => 2
[3,2,2] => [4,3] => 2
[3,2,1,1] => [3,2,1,1] => 1
[3,1,1,1,1] => [3,2,2] => 3
[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] => 2
[1,1,1,1,1,1,1] => [2,2,2,1] => 2
[8] => [1,1,1,1,1,1,1,1] => 1
[7,1] => [7,1] => 1
[6,2] => [3,3,1,1] => 1
[6,1,1] => [3,3,2] => 1
[5,3] => [5,3] => 1
[5,2,1] => [5,1,1,1] => 1
[5,1,1,1] => [5,2,1] => 1
[4,4] => [8] => 2
[4,3,1] => [3,1,1,1,1,1] => 1
[4,2,2] => [4,1,1,1,1] => 2
[4,2,1,1] => [2,1,1,1,1,1,1] => 2
[4,1,1,1,1] => [2,2,1,1,1,1] => 2
[3,3,2] => [6,1,1] => 2
[3,3,1,1] => [6,2] => 2
[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] => 2
[1,1,1,1,1,1,1,1] => [2,2,2,2] => 2
[9] => [9] => 1
[8,1] => [1,1,1,1,1,1,1,1,1] => 1
[7,2] => [7,1,1] => 1
[7,1,1] => [7,2] => 1
[6,3] => [3,3,3] => 1
[6,2,1] => [3,3,1,1,1] => 1
[6,1,1,1] => [3,3,2,1] => 1
[5,4] => [5,1,1,1,1] => 1
[5,3,1] => [5,3,1] => 1
[5,2,2] => [5,4] => 1
[5,2,1,1] => [5,2,1,1] => 1
[5,1,1,1,1] => [5,2,2] => 3
[4,4,1] => [8,1] => 2
[4,3,2] => [3,1,1,1,1,1,1] => 1
[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] => 2
[4,1,1,1,1,1] => [2,2,1,1,1,1,1] => 2
[3,3,3] => [6,3] => 2
[3,3,2,1] => [6,1,1,1] => 2
[3,3,1,1,1] => [6,2,1] => 2
[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] => 3
[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] => 2
[1,1,1,1,1,1,1,1,1] => [2,2,2,2,1] => 2
[10] => [5,5] => 1
[9,1] => [9,1] => 1
[8,2] => [1,1,1,1,1,1,1,1,1,1] => 1
[8,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[7,3] => [7,3] => 1
>>> Load all 287 entries. <<<
[7,2,1] => [7,1,1,1] => 1
[7,1,1,1] => [7,2,1] => 1
[6,4] => [3,3,1,1,1,1] => 1
[6,3,1] => [3,3,3,1] => 1
[6,2,2] => [4,3,3] => 2
[6,2,1,1] => [3,3,2,1,1] => 1
[6,1,1,1,1] => [3,3,2,2] => 1
[5,5] => [10] => 2
[5,4,1] => [5,1,1,1,1,1] => 1
[5,3,2] => [5,3,1,1] => 1
[5,3,1,1] => [5,3,2] => 1
[5,2,2,1] => [5,4,1] => 1
[5,2,1,1,1] => [5,2,1,1,1] => 1
[5,1,1,1,1,1] => [5,2,2,1] => 1
[4,4,2] => [8,1,1] => 2
[4,4,1,1] => [8,2] => 2
[4,3,3] => [6,1,1,1,1] => 2
[4,3,2,1] => [3,1,1,1,1,1,1,1] => 1
[4,3,1,1,1] => [3,2,1,1,1,1,1] => 1
[4,2,2,2] => [4,1,1,1,1,1,1] => 2
[4,2,2,1,1] => [4,2,1,1,1,1] => 2
[4,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[3,3,3,1] => [6,3,1] => 2
[3,3,2,2] => [6,4] => 2
[3,3,2,1,1] => [6,2,1,1] => 2
[3,3,1,1,1,1] => [6,2,2] => 2
[3,2,2,2,1] => [4,3,1,1,1] => 2
[3,2,2,1,1,1] => [4,3,2,1] => 2
[3,2,1,1,1,1,1] => [3,2,2,1,1,1] => 1
[3,1,1,1,1,1,1,1] => [3,2,2,2,1] => 3
[2,2,2,2,2] => [4,4,1,1] => 2
[2,2,2,2,1,1] => [4,4,2] => 2
[2,2,2,1,1,1,1] => [4,2,2,1,1] => 2
[2,2,1,1,1,1,1,1] => [4,2,2,2] => 2
[2,1,1,1,1,1,1,1,1] => [2,2,2,2,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2] => 2
[11] => [11] => 1
[10,1] => [5,5,1] => 1
[9,2] => [9,1,1] => 1
[9,1,1] => [9,2] => 1
[8,3] => [3,1,1,1,1,1,1,1,1] => 1
[8,2,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[8,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 2
[7,4] => [7,1,1,1,1] => 1
[7,3,1] => [7,3,1] => 1
[7,2,2] => [7,4] => 1
[7,2,1,1] => [7,2,1,1] => 1
[7,1,1,1,1] => [7,2,2] => 3
[6,5] => [5,3,3] => 1
[6,4,1] => [3,3,1,1,1,1,1] => 1
[6,3,2] => [3,3,3,1,1] => 1
[6,3,1,1] => [3,3,3,2] => 1
[6,2,2,1] => [4,3,3,1] => 2
[6,2,1,1,1] => [3,3,2,1,1,1] => 1
[6,1,1,1,1,1] => [3,3,2,2,1] => 1
[5,5,1] => [10,1] => 2
[5,4,2] => [5,1,1,1,1,1,1] => 1
[5,4,1,1] => [5,2,1,1,1,1] => 1
[5,3,3] => [6,5] => 2
[5,3,2,1] => [5,3,1,1,1] => 1
[5,3,1,1,1] => [5,3,2,1] => 1
[5,2,2,2] => [5,4,1,1] => 1
[5,2,2,1,1] => [5,4,2] => 3
[5,2,1,1,1,1] => [5,2,2,1,1] => 1
[5,1,1,1,1,1,1] => [5,2,2,2] => 3
[4,4,3] => [8,3] => 2
[4,4,2,1] => [8,1,1,1] => 2
[4,4,1,1,1] => [8,2,1] => 2
[4,3,3,1] => [6,1,1,1,1,1] => 2
[4,3,2,2] => [4,3,1,1,1,1] => 2
[4,3,2,1,1] => [3,2,1,1,1,1,1,1] => 1
[4,3,1,1,1,1] => [3,2,2,1,1,1,1] => 1
[4,2,2,2,1] => [4,1,1,1,1,1,1,1] => 2
[4,2,2,1,1,1] => [4,2,1,1,1,1,1] => 2
[4,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 2
[3,3,3,2] => [6,3,1,1] => 2
[3,3,3,1,1] => [6,3,2] => 2
[3,3,2,2,1] => [6,4,1] => 2
[3,3,2,1,1,1] => [6,2,1,1,1] => 2
[3,3,1,1,1,1,1] => [6,2,2,1] => 2
[3,2,2,2,2] => [4,4,3] => 2
[3,2,2,2,1,1] => [4,3,2,1,1] => 2
[3,2,2,1,1,1,1] => [4,3,2,2] => 2
[3,2,1,1,1,1,1,1] => [3,2,2,2,1,1] => 1
[3,1,1,1,1,1,1,1,1] => [3,2,2,2,2] => 3
[2,2,2,2,2,1] => [4,4,1,1,1] => 2
[2,2,2,2,1,1,1] => [4,4,2,1] => 2
[2,2,2,1,1,1,1,1] => [4,2,2,1,1,1] => 2
[2,2,1,1,1,1,1,1,1] => [4,2,2,2,1] => 2
[2,1,1,1,1,1,1,1,1,1] => [2,2,2,2,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2,1] => 2
[12] => [3,3,3,3] => 1
[11,1] => [11,1] => 1
[10,2] => [5,5,1,1] => 1
[10,1,1] => [5,5,2] => 1
[9,3] => [9,3] => 1
[9,2,1] => [9,1,1,1] => 1
[9,1,1,1] => [9,2,1] => 1
[8,4] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[8,3,1] => [3,1,1,1,1,1,1,1,1,1] => 1
[8,2,2] => [4,1,1,1,1,1,1,1,1] => 2
[8,2,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 2
[8,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2
[7,5] => [7,5] => 1
[7,4,1] => [7,1,1,1,1,1] => 1
[7,3,2] => [7,3,1,1] => 1
[7,3,1,1] => [7,3,2] => 1
[7,2,2,1] => [7,4,1] => 1
[7,2,1,1,1] => [7,2,1,1,1] => 1
[7,1,1,1,1,1] => [7,2,2,1] => 1
[6,6] => [12] => 2
[6,5,1] => [5,3,3,1] => 1
[6,4,2] => [3,3,1,1,1,1,1,1] => 1
[6,4,1,1] => [3,3,2,1,1,1,1] => 1
[6,3,3] => [6,3,3] => 2
[6,3,2,1] => [3,3,3,1,1,1] => 1
[6,3,1,1,1] => [3,3,3,2,1] => 1
[6,2,2,2] => [4,3,3,1,1] => 2
[6,2,2,1,1] => [4,3,3,2] => 2
[6,2,1,1,1,1] => [3,3,2,2,1,1] => 1
[6,1,1,1,1,1,1] => [3,3,2,2,2] => 2
[5,5,2] => [10,1,1] => 2
[5,5,1,1] => [10,2] => 2
[5,4,3] => [5,3,1,1,1,1] => 1
[5,4,2,1] => [5,1,1,1,1,1,1,1] => 1
[5,4,1,1,1] => [5,2,1,1,1,1,1] => 1
[5,3,3,1] => [6,5,1] => 2
[5,3,2,2] => [5,4,3] => 1
[5,3,2,1,1] => [5,3,2,1,1] => 1
[5,3,1,1,1,1] => [5,3,2,2] => 1
[5,2,2,2,1] => [5,4,1,1,1] => 1
[5,2,2,1,1,1] => [5,4,2,1] => 1
[5,2,1,1,1,1,1] => [5,2,2,1,1,1] => 1
[5,1,1,1,1,1,1,1] => [5,2,2,2,1] => 2
[4,4,4] => [8,1,1,1,1] => 2
[4,4,3,1] => [8,3,1] => 2
[4,4,2,2] => [8,4] => 2
[4,4,2,1,1] => [8,2,1,1] => 2
[4,4,1,1,1,1] => [8,2,2] => 2
[4,3,3,2] => [6,1,1,1,1,1,1] => 2
[4,3,3,1,1] => [6,2,1,1,1,1] => 2
[4,3,2,2,1] => [4,3,1,1,1,1,1] => 2
[4,3,2,1,1,1] => [3,2,1,1,1,1,1,1,1] => 1
[4,3,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 1
[4,2,2,2,2] => [4,4,1,1,1,1] => 2
[4,2,2,2,1,1] => [4,2,1,1,1,1,1,1] => 2
[4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => 2
[4,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1,1] => [2,2,2,2,1,1,1,1] => 2
[3,3,3,3] => [6,6] => 2
[3,3,3,2,1] => [6,3,1,1,1] => 2
[3,3,3,1,1,1] => [6,3,2,1] => 2
[3,3,2,2,2] => [6,4,1,1] => 2
[3,3,2,2,1,1] => [6,4,2] => 2
[3,3,2,1,1,1,1] => [6,2,2,1,1] => 2
[3,3,1,1,1,1,1,1] => [6,2,2,2] => 2
[3,2,2,2,2,1] => [4,4,3,1] => 2
[3,2,2,2,1,1,1] => [4,3,2,1,1,1] => 2
[3,2,2,1,1,1,1,1] => [4,3,2,2,1] => 2
[3,2,1,1,1,1,1,1,1] => [3,2,2,2,1,1,1] => 1
[3,1,1,1,1,1,1,1,1,1] => [3,2,2,2,2,1] => 3
[2,2,2,2,2,2] => [4,4,4] => 2
[2,2,2,2,2,1,1] => [4,4,2,1,1] => 2
[2,2,2,2,1,1,1,1] => [4,4,2,2] => 2
[2,2,2,1,1,1,1,1,1] => [4,2,2,2,1,1] => 2
[2,2,1,1,1,1,1,1,1,1] => [4,2,2,2,2] => 2
[2,1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [2,2,2,2,2,2] => 2
[6,5,2] => [5,3,3,1,1] => 1
[6,5,1,1] => [5,3,3,2] => 1
[6,2,2,1,1,1] => [4,3,3,2,1] => 2
[5,3,2,2,1] => [5,4,3,1] => 1
[5,3,1,1,1,1,1] => [5,3,2,2,1] => 1
[5,2,2,2,1,1] => [5,4,2,1,1] => 1
[5,2,2,1,1,1,1] => [5,4,2,2] => 3
[3,2,2,2,2,2] => [4,4,3,1,1] => 2
[3,2,2,2,2,1,1] => [4,4,3,2] => 2
[2,2,2,2,1,1,1,1,1] => [4,4,2,2,1] => 2
[6,5,1,1,1] => [5,3,3,2,1] => 1
[5,3,2,2,2] => [5,4,3,1,1] => 1
[5,3,2,2,1,1] => [5,4,3,2] => 1
[5,2,2,1,1,1,1,1] => [5,4,2,2,1] => 3
[3,2,2,2,2,1,1,1] => [4,4,3,2,1] => 2
[5,3,2,2,1,1,1] => [5,4,3,2,1] => 1
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Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
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].