Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s,ms\pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts these partitions, considering $d$-distinct allowing more general r)$-core partitions. As an application our approach, we obtain average maximum core

Abstra ct. Th e quantitat ive characterizat ion of complexity for several dynami c processes is approached by a mathematical scheme based on th e met ric and combinatori al prop ert ies of the space Z of finite measur able par titions . Precisely, Z is embed ded in a larger spa ce R of new objects, "rational part it ions," describin g the anti similarity between two prob abilistic experiments ....

PED partitions are with even parts distinct while odd unrestricted. Similarly, POD have In [16] several recurrence relations for the number of n proved analytically. They similar to relation given by Euler's pentagonal theorem. We provide combinatorial proofs all these theorems and also theorem analytically in [11], which motivated [16]. Moreover, we prove combinatorially a [7], Beck-type ident...

Every partition has, for some d, a Durfee square of side d. Every partition π with Durfee square of side d gives rise to a “successive rank vector” r = (r1, · · · , rd). Conversely, given a vector r = (r1, · · · , rd), there is a unique partition π0 of minimal size called the basis partition with r as its successive rank vector. We give a quick derivation of the generating function for b(n, d),...

Laboratory measurements were made of the sound insulations and absorption coefficients of a range of metal-framed partitions having built-in acoustic treatment. The partitions were made using glass wool and gypsum boards. The sound insulations of the single leaf partitions having built-in acoustic treatment were considerably lower than those of comparable partitions made using mineral wool, gyp...

In his memoir in 1984, George E. Andrews introduces many general classes of Frobenius partitions (simply F-partitions). Especially, he focuses his interest on two general classes of F-partitions: one is F-partitions that allow up to k repetitions of an integer in any row, and the other is F-partitions whose parts are taken from k copies of the nonnegative integers. The latter are called k color...

A method of Proctor [4] realizes the set of arbitrary plane partitions in a box and the set of symmetric plane partitions as bases of linear representations of Lie groups. We extend this method by realizing transposition and complementation of plane partitions as natural linear transformations of the representations, thereby enumerating symmetric plane partitions, self-complementary plane parti...

Non-Crossing Tree partitions are newer mathematical objects that have recent applications in genetics and mathematical biology. We explore several interesting connections between these partitions and the more commonly studied non-crossing set partitions. While non-crossing set partitions are counted by the Catalan numbers, we prove that non-crossing tree partitions in Binary trees are counted b...

In this paper we prove a bijection between 2-regular, k-noncrossing partitions and k-noncrossing enhanced partitions. Via this bijection we enumerate 2-regular, 3-noncrossing partitions using an enumeration result [1] for enhanced 3-noncrossing partitions. In addition we derive the asymptotics for the numbers of 2-regular, 3-noncrossing partitions using the BirkhoffTrijtzinky analytic theory of...

The definition and study of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations, begun with Klazar. Sagan continued his work by considering set partitions which avoids a single partition of three elements, and Goyt generalized these results by considering partitions which avoids any family of partitions of a 3-element set. In this paper we enumerate ...