Restricted k-color partitions

Abstract

Abstract. We generalize overpartitions to (k, j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. We find connections to divisor sums, the Han/Nekrasov-Okounkov hook length formula and a possible approach to a finitization, and other topics, suggesting that a rich mine of results is available.

Polynomial analogues of restricted multicolor b-ary partition functions

“…, where the subscripts denote the colors of the part, and by convention we write each partition in non-increasing order of its parts, and assign the colors also in nonincreasing order. This is consistent with the notation of colored ordinary partitions used by Keith [12]. Counting the partitions in this example, we see that C Λ 2 (4) = 13.…”

“…More recently, Rødseth and Sellers [16] introduced and studied b-ary overpartitions, in analogy to ordinary overpartitions that had been introduced a little earlier by Corteel and Lovejoy [4]. Ordinary overpartitions were extended to restricted multicolor partitions by Keith [12], and colored b-ary partitions have recently been studied by Ulas and Żmija [18].…”

Exact formulae for the fractional partition functions

“…When α ∈ Z + , p α (n) counts the number of partitions of n in which each term is labeled with one of α different colors, where the order of the colors does not matter [11]. Moreover, in such cases, the function η −α (τ ) = q − α 24 P (q) α (1.5)…”

Partitions into a small number of part sizes

“…This easily stated function has been studied by Major P. A. MacMahon [7], George Andrews [1], and more recently Tani and Bouroubi [10], the latter specifically interested in ν 2 . The author in a recent paper [5] stated several theorems concerning ν 2 and ventured further conjectures regarding ν 2 and ν 3 , which it is the purpose of this paper to prove and expand. Despite attention from these authors, results of the kind found in other areas of partition theory, such as congruences in arithmetic progressions, have not been forthcoming; here we provide several, with a proof strategy easily adaptable to future possible candidates.…”