A relationship is established between the factorization of 24n + 1 and the 5-divisibility of Q(n), where Q(n)is the number of partitions of n into distinct parts. As an application, an abundance of infinite families of congruences for Q(n) modulo powers of 5 are explicitly exhibited.[...
(n,k) := the number of partitions of n into k distinct parts.Also, we give lower and upper bounds for the density of the set {n ∈ N : p(n,k) ≡i(mod m)}, where m ≥ 2 and 0 ≤ i ≤ m − 1.Keywords: Restricted integer partitions; Restricted partition function.2010 MSC...
My approach for small n Letsmagic(left,last)magic(left,last)is the number of valid subsequences whose sum equalleftleftwhich next selected element is suchnextnextin range(last,left](last,left](nextnextis strictly greater then last selected numberlastlastand not greater than current sumleftleft)...
We prove new formulas and congruences for p(n,k):= the number of partitions of n into k parts and q(n,k):= the number of partitions of n into k distinct parts. Also, we give lower and upper bounds for the density of the set {n∈N:p(n,k)≡i(modm)}, where m≥2 and 0...
We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of partitions of i into distinct parts with even ...
Integer partitionsHecke eigenformsLet p e d ( n ) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for p e d ( n ) mod 2 and mod 4 by the theory of Hecke eigenforms....
gaveanelementaryof(1.3).(1.4)wasprovedbyA.O.L.Atkin[2]in1967.Afterthat,the generalizationsof(1.2),(1.3)and(1.4)forotherpartitionfunctionshavebeeninvestigated bymanymathematicians.Letq(n)denotethenumberofpartitionsofnintodistinctparts, GordonandHughes[10]obtainedsomecongruencesforq(n)modulopowers5.They...
n−1 x n . 2 Chapter 1. Compositions and Partitions Examples. As an exercise I would suggest using both the combinatorial method and the algebraic approach to prove the following results: (1) The number of compositions of n into exactly m parts is n −1 m−1 (Catalan); (2) The...
作者: N Kravitz 摘要: We investigate the number $N_{d,r}(s)$ of $(s, s+r)$-core integer partitions with $d$-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics, and exact formulas...
A natural example of such a function is the A-partition function pA(n), which enumerates the number of partitions of n with parts in the fixed finite multiset A={a1,a2,…,ak} of positive integers. For an arbitrary positive integer d, we present efficient criteria for both the order d...