Identifier
Identifier
Values
[1] generating graphics... => 1
[2] generating graphics... => 2
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 1
[4] generating graphics... => 2
[3,1] generating graphics... => 1
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => 2
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 1
[4,1] generating graphics... => 2
[3,2] generating graphics... => 1
[3,1,1] generating graphics... => 1
[2,2,1] generating graphics... => 2
[2,1,1,1] generating graphics... => 2
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 2
[5,1] generating graphics... => 1
[4,2] generating graphics... => 2
[4,1,1] generating graphics... => 2
[3,3] generating graphics... => 1
[3,2,1] generating graphics... => 1
[3,1,1,1] generating graphics... => 1
[2,2,2] generating graphics... => 2
[2,2,1,1] generating graphics... => 2
[2,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1
[6,1] generating graphics... => 2
[5,2] generating graphics... => 1
[5,1,1] generating graphics... => 1
[4,3] generating graphics... => 2
[4,2,1] generating graphics... => 2
[4,1,1,1] generating graphics... => 2
[3,3,1] generating graphics... => 1
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => 1
[3,1,1,1,1] generating graphics... => 1
[2,2,2,1] generating graphics... => 2
[2,2,1,1,1] generating graphics... => 2
[2,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 2
[7,1] generating graphics... => 1
[6,2] generating graphics... => 2
[6,1,1] generating graphics... => 2
[5,3] generating graphics... => 1
[5,2,1] generating graphics... => 1
[5,1,1,1] generating graphics... => 1
[4,4] generating graphics... => 2
[4,3,1] generating graphics... => 2
[4,2,2] generating graphics... => 2
[4,2,1,1] generating graphics... => 2
[4,1,1,1,1] generating graphics... => 2
[3,3,2] generating graphics... => 1
[3,3,1,1] generating graphics... => 1
[3,2,2,1] generating graphics... => 1
[3,2,1,1,1] generating graphics... => 1
[3,1,1,1,1,1] generating graphics... => 1
[2,2,2,2] generating graphics... => 2
[2,2,2,1,1] generating graphics... => 2
[2,2,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 1
[8,1] generating graphics... => 2
[7,2] generating graphics... => 1
[7,1,1] generating graphics... => 1
[6,3] generating graphics... => 2
[6,2,1] generating graphics... => 2
[6,1,1,1] generating graphics... => 2
[5,4] generating graphics... => 1
[5,3,1] generating graphics... => 1
[5,2,2] generating graphics... => 3
[5,2,1,1] generating graphics... => 1
[5,1,1,1,1] generating graphics... => 1
[4,4,1] generating graphics... => 2
[4,3,2] generating graphics... => 2
[4,3,1,1] generating graphics... => 2
[4,2,2,1] generating graphics... => 2
[4,2,1,1,1] generating graphics... => 2
[4,1,1,1,1,1] generating graphics... => 2
[3,3,3] generating graphics... => 1
[3,3,2,1] generating graphics... => 1
[3,3,1,1,1] generating graphics... => 1
[3,2,2,2] generating graphics... => 3
[3,2,2,1,1] generating graphics... => 1
[3,2,1,1,1,1] generating graphics... => 1
[3,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,1] generating graphics... => 2
[2,2,2,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 2
[9,1] generating graphics... => 1
[8,2] generating graphics... => 2
[8,1,1] generating graphics... => 2
[7,3] generating graphics... => 1
[7,2,1] generating graphics... => 1
[7,1,1,1] generating graphics... => 1
[6,4] generating graphics... => 2
[6,3,1] generating graphics... => 2
[6,2,2] generating graphics... => 2
[6,2,1,1] generating graphics... => 2
[6,1,1,1,1] generating graphics... => 2
[5,5] generating graphics... => 1
[5,4,1] generating graphics... => 1
[5,3,2] generating graphics... => 1
[5,3,1,1] generating graphics... => 1
[5,2,2,1] generating graphics... => 1
[5,2,1,1,1] generating graphics... => 1
[5,1,1,1,1,1] generating graphics... => 1
[4,4,2] generating graphics... => 2
[4,4,1,1] generating graphics... => 2
[4,3,3] generating graphics... => 2
[4,3,2,1] generating graphics... => 2
[4,3,1,1,1] generating graphics... => 2
[4,2,2,2] generating graphics... => 2
[4,2,2,1,1] generating graphics... => 2
[4,2,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,1] generating graphics... => 1
[3,3,2,2] generating graphics... => 1
[3,3,2,1,1] generating graphics... => 1
[3,3,1,1,1,1] generating graphics... => 1
[3,2,2,2,1] generating graphics... => 3
[3,2,2,1,1,1] generating graphics... => 1
[3,2,1,1,1,1,1] generating graphics... => 1
[3,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2] generating graphics... => 2
[2,2,2,2,1,1] generating graphics... => 2
[2,2,2,1,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 1
[10,1] generating graphics... => 2
[9,2] generating graphics... => 1
[9,1,1] generating graphics... => 1
[8,3] generating graphics... => 2
[8,2,1] generating graphics... => 2
[8,1,1,1] generating graphics... => 2
[7,4] generating graphics... => 1
[7,3,1] generating graphics... => 1
[7,2,2] generating graphics... => 3
[7,2,1,1] generating graphics... => 1
[7,1,1,1,1] generating graphics... => 1
[6,5] generating graphics... => 2
[6,4,1] generating graphics... => 2
[6,3,2] generating graphics... => 2
[6,3,1,1] generating graphics... => 2
[6,2,2,1] generating graphics... => 2
[6,2,1,1,1] generating graphics... => 2
[6,1,1,1,1,1] generating graphics... => 2
[5,5,1] generating graphics... => 1
[5,4,2] generating graphics... => 3
[5,4,1,1] generating graphics... => 1
[5,3,3] generating graphics... => 1
[5,3,2,1] generating graphics... => 1
[5,3,1,1,1] generating graphics... => 1
[5,2,2,2] generating graphics... => 3
[5,2,2,1,1] generating graphics... => 1
[5,2,1,1,1,1] generating graphics... => 1
[5,1,1,1,1,1,1] generating graphics... => 1
[4,4,3] generating graphics... => 2
[4,4,2,1] generating graphics... => 2
[4,4,1,1,1] generating graphics... => 2
[4,3,3,1] generating graphics... => 2
[4,3,2,2] generating graphics... => 2
[4,3,2,1,1] generating graphics... => 2
[4,3,1,1,1,1] generating graphics... => 2
[4,2,2,2,1] generating graphics... => 2
[4,2,2,1,1,1] generating graphics... => 2
[4,2,1,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,2] generating graphics... => 1
[3,3,3,1,1] generating graphics... => 1
[3,3,2,2,1] generating graphics... => 1
[3,3,2,1,1,1] generating graphics... => 1
[3,3,1,1,1,1,1] generating graphics... => 1
[3,2,2,2,2] generating graphics... => 3
[3,2,2,2,1,1] generating graphics... => 1
[3,2,2,1,1,1,1] generating graphics... => 1
[3,2,1,1,1,1,1,1] generating graphics... => 1
[3,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,1] generating graphics... => 2
[2,2,2,2,1,1,1] generating graphics... => 2
[2,2,2,1,1,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 2
[11,1] generating graphics... => 1
[10,2] generating graphics... => 2
[10,1,1] generating graphics... => 2
[9,3] generating graphics... => 1
[9,2,1] generating graphics... => 1
[9,1,1,1] generating graphics... => 1
[8,4] generating graphics... => 2
[8,3,1] generating graphics... => 2
[8,2,2] generating graphics... => 2
[8,2,1,1] generating graphics... => 2
[8,1,1,1,1] generating graphics... => 2
[7,5] generating graphics... => 1
[7,4,1] generating graphics... => 1
[7,3,2] generating graphics... => 1
[7,3,1,1] generating graphics... => 1
[7,2,2,1] generating graphics... => 1
[7,2,1,1,1] generating graphics... => 1
[7,1,1,1,1,1] generating graphics... => 1
[6,6] generating graphics... => 2
[6,5,1] generating graphics... => 2
[6,4,2] generating graphics... => 2
[6,4,1,1] generating graphics... => 2
[6,3,3] generating graphics... => 2
[6,3,2,1] generating graphics... => 2
[6,3,1,1,1] generating graphics... => 2
[6,2,2,2] generating graphics... => 2
[6,2,2,1,1] generating graphics... => 2
[6,2,1,1,1,1] generating graphics... => 2
[6,1,1,1,1,1,1] generating graphics... => 2
[5,5,2] generating graphics... => 1
[5,5,1,1] generating graphics... => 1
[5,4,3] generating graphics... => 1
[5,4,2,1] generating graphics... => 1
[5,4,1,1,1] generating graphics... => 1
[5,3,3,1] generating graphics... => 1
[5,3,2,2] generating graphics... => 1
[5,3,2,1,1] generating graphics... => 1
[5,3,1,1,1,1] generating graphics... => 1
[5,2,2,2,1] generating graphics... => 2
[5,2,2,1,1,1] generating graphics... => 1
[5,2,1,1,1,1,1] generating graphics... => 1
[5,1,1,1,1,1,1,1] generating graphics... => 1
[4,4,4] generating graphics... => 2
[4,4,3,1] generating graphics... => 2
[4,4,2,2] generating graphics... => 2
[4,4,2,1,1] generating graphics... => 2
[4,4,1,1,1,1] generating graphics... => 2
[4,3,3,2] generating graphics... => 2
[4,3,3,1,1] generating graphics... => 2
[4,3,2,2,1] generating graphics... => 2
[4,3,2,1,1,1] generating graphics... => 2
[4,3,1,1,1,1,1] generating graphics... => 2
[4,2,2,2,2] generating graphics... => 2
[4,2,2,2,1,1] generating graphics... => 2
[4,2,2,1,1,1,1] generating graphics... => 2
[4,2,1,1,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,3] generating graphics... => 1
[3,3,3,2,1] generating graphics... => 1
[3,3,3,1,1,1] generating graphics... => 1
[3,3,2,2,2] generating graphics... => 2
[3,3,2,2,1,1] generating graphics... => 1
[3,3,2,1,1,1,1] generating graphics... => 1
[3,3,1,1,1,1,1,1] generating graphics... => 1
[3,2,2,2,2,1] generating graphics... => 3
[3,2,2,2,1,1,1] generating graphics... => 1
[3,2,2,1,1,1,1,1] generating graphics... => 1
[3,2,1,1,1,1,1,1,1] generating graphics... => 1
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,2] generating graphics... => 2
[2,2,2,2,2,1,1] generating graphics... => 2
[2,2,2,2,1,1,1,1] generating graphics... => 2
[2,2,2,1,1,1,1,1,1] generating graphics... => 2
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 2
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[5,4,3,1] generating graphics... => 1
[5,4,2,2] generating graphics... => 3
[5,4,2,1,1] generating graphics... => 1
[5,3,3,2] generating graphics... => 1
[5,3,3,1,1] generating graphics... => 1
[5,3,2,2,1] generating graphics... => 1
[4,4,3,2] generating graphics... => 2
[4,4,3,1,1] generating graphics... => 2
[4,4,2,2,1] generating graphics... => 2
[4,3,3,2,1] generating graphics... => 2
[5,4,3,2] generating graphics... => 1
[5,4,3,1,1] generating graphics... => 1
[5,4,2,2,1] generating graphics... => 3
[5,3,3,2,1] generating graphics... => 1
[4,4,3,2,1] generating graphics... => 2
[5,4,3,2,1] generating graphics... => 1
click to show generating function       
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.
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