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Augmented monomials in terms of power sums
No full textAny positive integer n can be written as a sum of one or more positive integers, i.e., When the order of integers i does not matter, this representation is known as an integer partition Andrews (1976) and can be rewritten as where each positive integer i appears ti times. If the order of integers i is important, then the representation (1) is known as a composition. For we have a descending composition. We notice that more often than not there appears the tendency of defining partitions as descending compositions and this is also the con-vention used in this paper. In order to indicate that is a partition of n, we use the notation ⊢ n. We denote by l() the number of parts of , i.e., (1)n = 1 + 2 + · · · + r. n = t1 + 2t2 + · · · + ntn 1 2 · · · r = [1, 2,..., r] or = [1 t12t2... ntn] l() = r or l() = t1 + t2 + · · · + tn. The problem of base changes for the classical symmetric functions has been solved a long time ago and has been incorporated into most computer software packages for symmetric functions. In this paper, we develop a simple recursive formula for the expansion of the augmented monomial symmetric functions into power sum symmet-ric functions. As corollaries, we present two algorithms that can be used to expressing the augmented monomial symmetric functions in terms of the power sum symmetric functions
An infinite sequence of inequalities involving special values of the Riemann zeta function
No full textA <i>q</i>-Series Congruence Inspired by Andrews and Ramanujan
No full textFor each s∈{1,3,5}, we consider Rs(n) to be the number of the partitions of n into parts not congruent to 0, ±s(mod12). In recent years, some relations for computing the value of R3(n) were studied. In this paper, we investigate the parity of Rs(n) when s∈{1,5} and derive the following congruence identity: ∑n=1∞(−q;q)n−12(1+qn)qn2(q;q)2n≡∑n=1∞qn2+q3n2(mod2). For each s∈{1,5}, the number of the partitions of n into parts not congruent to 0, ±s(mod12) is connected with two truncated theta series. Some open problems involving R1(n) and R5(n) are introduced in this context
Plane Partitions and a Problem of Josephus
No full textThe Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the position L(n,k) in which you should stand in order to be the last survivor. Let Jn be the order in which the first person is executed on counting when k=2. In this paper, we consider the sequence (Jn)n⩾1 in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving Jn. Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors’ functions and Jn are provided in this context
An infinite sequence of inequalities involving special values of the Riemann zeta function
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