Partitions and overpartitions with attached parts

We show how to interpret a certain q-series as a generating function for overpartitions with attached parts. A number of families of partition theorems follow as corollaries. Received: 12 April 2006     

Partitions with parts in a finite set

Let be a nonempty finite set of relatively prime positive integers, and let denote the number of partitions of with parts in . An elementary arithmetic argument is used to prove the asymptotic formula

     

Partitions of bi-partite numbers into at mostj parts

The number of partitions of a bi-partite number into at mostj parts is studied. We consider this function,p _j (x, y), on the linex+y=2n. Forj4, we show that this function is maximized whenx=y. Forj>4 we provide an explicit formula forn _j so that, for allnn _j ,x=y yields a maximum forp _j (x,y).     

Partitions with initial repetitions

A variety of interesting connections with modular forms, mock theta functions and Rogers- Ramanujan type identities arise in consideration of partitions in which the smaller integers are repeated as summands more often than the larger summands. In particular, this concept leads to new interpretations of the Rogers Selberg identities and Bailey＇s modulus 9 identities.     

Partitions with k Crossings

We study the number S_k(n) of partitions of the set {1, 2,..., n} with k crossings and show that for each k, their ordinary generating function S_k(x) is a rational function of x and the ordinary generating function of the Catalan numbers. If k = 1, then we get a sequence first found by Cayley in 1890.     

Partitions with prescribed mean curvatures

We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001     

Partitions with certain intersection properties

Partitions of the n ‐element set are considered. A family of m such partitions is called an ( n, m, k )‐pamily, if there are two classes for any pair of partitions whose intersection has at least k elements, and any pair of elements is in the same class for at most two partitions. Let f ( n, k ) denote the maximum of m for which an ( n, m, k )‐pamily exist. A constructive lower bound is given for f ( n, k ), which is compared with the trivial upper bound. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:345‐354, 2011     

Partitions and sums of integers with repetition

A partition of N is called “admissible” provided some cell has arbitrarily long arithmetic progressions of even integers in a fixed increment. The principal result is that the statement “Whenever {A_i}_{i < r} is an admissible partition of N, there are some i < r and some sequence 〈x_n〉_{n < ω} of distinct members of N such that x_n + x_m?A_i whenever {m, n} ? ω″ is true when r = 2 and false when r ? 3.     

Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.     

Partitions and sums with inverses in Abelian groups

Let G be an Abelian group and let be infinite. We construct a partition of A such that whenever (xn)_n<ω is a one-to-one sequence in A, g∈G and m<ω, one has

(g+FSI((xn)_n<ω))∩Am≠∅,     

Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Göllnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.     

Optimized Local Trigonometric Bases with Nonuniform Partitions

The authors provide optimized local trigonometric bases with nonuniform partitions which efficiently compress trigonometric functions. Numerical examples demonstrate that in many cases the proposed bases provide better compression than the optimized bases with uniform partitions obtained by Matviyenko.     

Reciprocal Function Series Coefficients with Integer Partitions

We obtain an explicit formula in terms of the partitions of the positive integer n to express the nth coefficient of the formal series expansion of the reciprocal of a given function. A brief survey shows that our arithmetic proof differs from others, some obtained already in the XIX century. Examples are given to establish explicit formulas for Bernoulli, Euler, and Fibonacci numbers.