TY - JOUR

T1 - On the Unimodality of some Partition Polynomials

AU - Odlyzko, A. M.

AU - Richmond, L. B.

PY - 1982

Y1 - 1982

N2 - It is shown that for a wide class of sequences {ai} of positive integers, the polynomials Π i=1n(1+x ai)=∑ k=0N bkxk,N=∑ i=1n ai are almost unimodal; i.e. there is a constant A, independent of n, such that bA⩽bA+1⩽⋯⩽bK and bK⩾bK+1⩾⋯⩾bN-A, where K=N/2 or (N+1)/2. Among the sequences {ai} to which this result applies are the sequences given by ak=f(k), where f(x) is a polynomial with integer coefficients such that for every prime p there is an integer k with p∤f(k).

AB - It is shown that for a wide class of sequences {ai} of positive integers, the polynomials Π i=1n(1+x ai)=∑ k=0N bkxk,N=∑ i=1n ai are almost unimodal; i.e. there is a constant A, independent of n, such that bA⩽bA+1⩽⋯⩽bK and bK⩾bK+1⩾⋯⩾bN-A, where K=N/2 or (N+1)/2. Among the sequences {ai} to which this result applies are the sequences given by ak=f(k), where f(x) is a polynomial with integer coefficients such that for every prime p there is an integer k with p∤f(k).

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U2 - 10.1016/S0195-6698(82)80010-3

DO - 10.1016/S0195-6698(82)80010-3

M3 - Article

AN - SCOPUS:0040574141

SN - 0195-6698

VL - 3

SP - 69

EP - 84

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 1

ER -