11 papers
Neurosymbolic Learning of Interpretable World Models
Riccardo Andreoni, Massimiliano Ronzani, Chiara Ghidini +2
World models aim to learn compact representations of an environment from observations, enabling agents to predict future states and plan their behaviour. However, most existing app…
Further results on non-negativity conjectures for copartition products
Haijun Li
Burson and Eichhorn proposed infinite and finite coefficientwise non-negativity conjectures for products arising from copartitions. We obtain several further results. First, both p…
Proofs of five conjectural identities on modular rank four Nahm sums
Haijun Li
Nahm sums and Rogers-Ramanujan type identities have attracted considerable attention in recent years. In this paper, we provide analytic proofs of five conjectural identities on mo…
Linked partition ideals and gap-frequency partitions
Haijun Li
Recently, linked partition ideals have attracted renewed attention. In this paper, we embed the gap-frequency partitions in which each part appears at most twice or three times int…
On mesh patterns of short length: Equidistribution and enumeration
Qi Fang, Shishuo Fu, Sergey Kitaev +3
The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involutio…
Combinatorial perspectives on identities for partitions with distinct even parts
Haijun Li
Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpo…
Parity patterns meet Genocchi numbers, I: four labelings and three bijections
Quan Yuan, Qi Fang, Shishuo Fu +1
Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of…
On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings
Qi Fang, Shishuo Fu, Sergey Kitaev +1
Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures o…
Consecutive and quasi-consecutive patterns: -Wilf classifications and generating functions
Yan Wang, Qi Fang, Shishuo Fu +2
Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as e…
Sequences of odd length in strict partitions II: the -measure and refinements of Euler's theorem
Shishuo Fu, Haijun Li
The number of sequences of odd length in strict partitions (denoted as ), which plays a pivotal role in the first paper of this series, is investigated in different c…
Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities
Shishuo Fu, Haijun Li
Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a doubl…