8 papers
An optimal refinement-compatible bijection between singleton-free partitions and partitions without cyclic adjacencies
Vuong Bui
It is well known that the number of partitions of without singletons equals the number of partitions of in which no block contains two cyclically adjacent elements $i,i…
A direct injection for the strong -log-convexity of Touchard polynomials
Vuong Bui
We provide a direct injection for the well-known strong log-convexity of the Bell numbers , that is for every . Our injection $Î _m\tim…
A sharper log-convexity inequality for Bell numbers
Vuong Bui
We prove a stronger version of the log-convexity inequality for the Bell numbers . In particular, for , we have \[ B_{n+1}B_{n-1} - (B_n)^2 \ge \sum_{i=1}^{n} F_i (B_{…
The mapping index through the lens of the cross-index
Vuong Bui, Hamid Reza Daneshpajouh, Roman Karasev
We study the cross-index of free \(G\)-posets as a combinatorial analogue of the equivariant topological index. We demonstrate that the cross-index exhibits many structural propert…
A convolutional approach to bounding the number of polyominoes
Vuong Bui
Although known lower bounds for the growth rate of polyominoes, or Klarner's constant, are already close to the empirically estimated value , almost no conceptual progre…
A short proof of an upper bound on the growth constant of polyiamonds
Vuong Bui
We provide a short and elementary proof that the growth constant of polyiamonds is at most for the unique real root of the equation . This coincidenta…