On a conjecture of Widom

On the Widom-Rowlinson Occupancy Fraction in Regular Graphs

We consider the Widom-Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d+1 vertices, $K_{d+1}$'s. As a corollary we find that $K_{d+1}$ also maximises the normalised partition function of the Widom-Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalised number of homomorphisms from any d-regular graph…

On some conjectures of P. Barry

We prove a number of conjectures [arXiv:2005.04066] recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence.

A proof of the compositional Delta conjecture

We prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson (2018) for $Δ_{e_{n-k-1}}'e_n$ which was stated by D'Adderio, Iraci and Vanden Wyngaerd (2020) in terms of Theta operators.

关于全正矩阵子式的有界比率

On bounded ratios of minors of totally positive matrices

本文通过提供无法分解为原始比率乘积的全正矩阵子式有界Laurent单项式实例，否定了Fallat等人的分解猜想，但验证了减法自由猜想，并证明所有有界比率构成特定维度的多面体锥。

On Kalai's conjectures concerning centrally symmetric polytopes

In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \geq 5.

关于Schmidt与Tuller一维装填与覆盖猜想的解答

Solution to a conjecture of Schmidt and Tuller on one-dimensional packings and coverings

本研究证实了Schmidt与Tuller在2008年提出的关于三点集平移的整数最优装填与覆盖猜想，并将该猜想与组合数学中的多个其他问题相关联。