edit this statistic or download as text // json
Identifier
Values
=>
Cc0002;cc-rep
[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 [7,2,1]=>1 [7,1,1,1]=>1 [6,4]=>2 [6,3,1]=>2 [6,2,2]=>2 [6,2,1,1]=>2 [6,1,1,1,1]=>2 [5,5]=>1 [5,4,1]=>1 [5,3,2]=>1 [5,3,1,1]=>1 [5,2,2,1]=>1 [5,2,1,1,1]=>1 [5,1,1,1,1,1]=>1 [4,4,2]=>2 [4,4,1,1]=>2 [4,3,3]=>2 [4,3,2,1]=>2 [4,3,1,1,1]=>2 [4,2,2,2]=>2 [4,2,2,1,1]=>2 [4,2,1,1,1,1]=>2 [4,1,1,1,1,1,1]=>2 [3,3,3,1]=>1 [3,3,2,2]=>1 [3,3,2,1,1]=>1 [3,3,1,1,1,1]=>1 [3,2,2,2,1]=>3 [3,2,2,1,1,1]=>1 [3,2,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1]=>1 [2,2,2,2,2]=>2 [2,2,2,2,1,1]=>2 [2,2,2,1,1,1,1]=>2 [2,2,1,1,1,1,1,1]=>2 [2,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1]=>1 [11]=>1 [10,1]=>2 [9,2]=>1 [9,1,1]=>1 [8,3]=>2 [8,2,1]=>2 [8,1,1,1]=>2 [7,4]=>1 [7,3,1]=>1 [7,2,2]=>3 [7,2,1,1]=>1 [7,1,1,1,1]=>1 [6,5]=>2 [6,4,1]=>2 [6,3,2]=>2 [6,3,1,1]=>2 [6,2,2,1]=>2 [6,2,1,1,1]=>2 [6,1,1,1,1,1]=>2 [5,5,1]=>1 [5,4,2]=>3 [5,4,1,1]=>1 [5,3,3]=>1 [5,3,2,1]=>1 [5,3,1,1,1]=>1 [5,2,2,2]=>3 [5,2,2,1,1]=>1 [5,2,1,1,1,1]=>1 [5,1,1,1,1,1,1]=>1 [4,4,3]=>2 [4,4,2,1]=>2 [4,4,1,1,1]=>2 [4,3,3,1]=>2 [4,3,2,2]=>2 [4,3,2,1,1]=>2 [4,3,1,1,1,1]=>2 [4,2,2,2,1]=>2 [4,2,2,1,1,1]=>2 [4,2,1,1,1,1,1]=>2 [4,1,1,1,1,1,1,1]=>2 [3,3,3,2]=>1 [3,3,3,1,1]=>1 [3,3,2,2,1]=>1 [3,3,2,1,1,1]=>1 [3,3,1,1,1,1,1]=>1 [3,2,2,2,2]=>3 [3,2,2,2,1,1]=>1 [3,2,2,1,1,1,1]=>1 [3,2,1,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1,1]=>1 [2,2,2,2,2,1]=>2 [2,2,2,2,1,1,1]=>2 [2,2,2,1,1,1,1,1]=>2 [2,2,1,1,1,1,1,1,1]=>2 [2,1,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1,1]=>1 [12]=>2 [11,1]=>1 [10,2]=>2 [10,1,1]=>2 [9,3]=>1 [9,2,1]=>1 [9,1,1,1]=>1 [8,4]=>2 [8,3,1]=>2 [8,2,2]=>2 [8,2,1,1]=>2 [8,1,1,1,1]=>2 [7,5]=>1 [7,4,1]=>1 [7,3,2]=>1 [7,3,1,1]=>1 [7,2,2,1]=>1 [7,2,1,1,1]=>1 [7,1,1,1,1,1]=>1 [6,6]=>2 [6,5,1]=>2 [6,4,2]=>2 [6,4,1,1]=>2 [6,3,3]=>2 [6,3,2,1]=>2 [6,3,1,1,1]=>2 [6,2,2,2]=>2 [6,2,2,1,1]=>2 [6,2,1,1,1,1]=>2 [6,1,1,1,1,1,1]=>2 [5,5,2]=>1 [5,5,1,1]=>1 [5,4,3]=>1 [5,4,2,1]=>1 [5,4,1,1,1]=>1 [5,3,3,1]=>1 [5,3,2,2]=>1 [5,3,2,1,1]=>1 [5,3,1,1,1,1]=>1 [5,2,2,2,1]=>2 [5,2,2,1,1,1]=>1 [5,2,1,1,1,1,1]=>1 [5,1,1,1,1,1,1,1]=>1 [4,4,4]=>2 [4,4,3,1]=>2 [4,4,2,2]=>2 [4,4,2,1,1]=>2 [4,4,1,1,1,1]=>2 [4,3,3,2]=>2 [4,3,3,1,1]=>2 [4,3,2,2,1]=>2 [4,3,2,1,1,1]=>2 [4,3,1,1,1,1,1]=>2 [4,2,2,2,2]=>2 [4,2,2,2,1,1]=>2 [4,2,2,1,1,1,1]=>2 [4,2,1,1,1,1,1,1]=>2 [4,1,1,1,1,1,1,1,1]=>2 [3,3,3,3]=>1 [3,3,3,2,1]=>1 [3,3,3,1,1,1]=>1 [3,3,2,2,2]=>2 [3,3,2,2,1,1]=>1 [3,3,2,1,1,1,1]=>1 [3,3,1,1,1,1,1,1]=>1 [3,2,2,2,2,1]=>3 [3,2,2,2,1,1,1]=>1 [3,2,2,1,1,1,1,1]=>1 [3,2,1,1,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1,1,1]=>1 [2,2,2,2,2,2]=>2 [2,2,2,2,2,1,1]=>2 [2,2,2,2,1,1,1,1]=>2 [2,2,2,1,1,1,1,1,1]=>2 [2,2,1,1,1,1,1,1,1,1]=>2 [2,1,1,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1,1,1]=>1 [5,4,3,1]=>1 [5,4,2,2]=>3 [5,4,2,1,1]=>1 [5,3,3,2]=>1 [5,3,3,1,1]=>1 [5,3,2,2,1]=>1 [4,4,3,2]=>2 [4,4,3,1,1]=>2 [4,4,2,2,1]=>2 [4,3,3,2,1]=>2 [5,4,3,2]=>1 [5,4,3,1,1]=>1 [5,4,2,2,1]=>3 [5,3,3,2,1]=>1 [4,4,3,2,1]=>2 [5,4,3,2,1]=>1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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