Weighted cylindric partitions

Abstract

Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double-shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the Göllnitz–Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.

The factorial-basis method for finding definite-sum solutions of linear recurrences with polynomial coefficients

“…This is enough to deduce that ′ = (− ) . These types of series and recurrences are common in -series and partitions [5,6,8,22,23]. In fact, we constructed the simple example, Example 5.6, by looking at a recent paper of Andrewsvan Ekeren-Heluani [5, (3.4.1)].…”

Combinatorial Constructions of Generating Functions of Cylindric Partitions with Small Profiles into Unrestricted or Distinct Parts

“…(1) j+c 1 . For example, the sequence Λ = ( (6,5,4,4), (8,8,5,3), (7,6,4,2)) is a cylindric partition with profile (1, 2, 0). One can check that for all j, λ We can visualize the required inequalities by writing the partitions in subsequent rows repeating the first row below the last one, and shifting the rows below as much as necessary to the left.…”

“…We construct generating functions for such partitions with profiles c = (1, 1) or c = (2, 0), which turn out to be combinations of infinite products. We refer the reader to [4], where cylindric partitions into distinct parts are also studied. We conclude by constructing an evidently positive series generating function for cylindric partitions with small profiles into odd parts.…”

“…We read the parts of λ as pairs: [6,7], [5,4], [4,4] and [0, 3]. We now start to perform the moves defined in the proof of Theorem 3.…”

“…We first change the places of 0 and 3 in the rightmost pair and we get the following intermediate partition: We changed the total weight by 7, so we have β 1 = 7. We do not touch to the pairs [3,3] and [4,3] since 3 3 and 4 3. We now correct the places of 6 and 5, then we perform the last possible move: We changed the total weight by 1, so we have β 2 = 1.…”