- FindStat

Identifier

St000755: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 2

[1,1] => 1

[3] => 1

[2,1] => 2

[1,1,1] => 1

[4] => 2

[3,1] => 1

[2,2] => 2

[2,1,1] => 2

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

[5] => 1

[4,1] => 2

[3,2] => 1

[3,1,1] => 1

[2,2,1] => 2

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

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

[6] => 2

[5,1] => 1

[4,2] => 2

[4,1,1] => 2

[3,3] => 1

[3,2,1] => 1

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

[2,2,2] => 2

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

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

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

[7] => 1

[6,1] => 2

[5,2] => 1

[5,1,1] => 1

[4,3] => 2

[4,2,1] => 2

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

[3,3,1] => 1

[3,2,2] => 3

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

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

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

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

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

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

[8] => 2

[7,1] => 1

[6,2] => 2

[6,1,1] => 2

[5,3] => 1

[5,2,1] => 1

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

[4,4] => 2

[4,3,1] => 2

[4,2,2] => 2

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

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

[3,3,2] => 1

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

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

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

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

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

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

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

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

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

[9] => 1

[8,1] => 2

[7,2] => 1

[7,1,1] => 1

[6,3] => 2

[6,2,1] => 2

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

[5,4] => 1

[5,3,1] => 1

[5,2,2] => 3

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

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

[4,4,1] => 2

[4,3,2] => 2

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

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

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

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

[3,3,3] => 1

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

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

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

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

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

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

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

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

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

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

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

[10] => 2

[9,1] => 1

[8,2] => 2

[8,1,1] => 2

[7,3] => 1

>>> Load all 287 entries. <<<

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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.

References

[1] wikipedia:Recurrence_relation#Roots_of_the_characteristic_polynomial

Code

def statistic(m):
    """
    Return the number of real roots of

    x^m_k = x^{m_k-m_1} + x^{m_k-m_2} + ... + 1

    without multiplicities.
    """
    if len(m) == 0:
        return None
    R. = PolynomialRing(ZZ)
    mk = max(m)
    eq = x^mk - sum(x^(mk-e) for e in m)
    return eq.number_of_real_roots()

Created

Apr 08, 2017 at 16:43 by Martin Rubey

Updated

Dec 30, 2017 at 22:57 by Martin Rubey