Koroljuk's formula for counting lattice paths revisited

Counting extensions revisited

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguin…

The Kirchhoff's Matrix-Tree Theorem revisited: counting spanning trees with the quantum relative entropy

By revisiting the Kirchhoff's Matrix-Tree Theorem, we give an exact formula for the number of spanning trees of a graph in terms of the quantum relative entropy between the maximally mixed state and another state specifically obtained from the graph. We use properties of the quantum relative entropy to prove tight bounds for the number of spanning trees in terms of basic parameters like degrees and number of vertices.

P-partitions revisited

We compare a traditional and non-traditional view on the subject of P-partitions, leading to formulas counting linear extensions of certain posets.

1324-avoiding permutations revisited

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in $14$ further terms of the generating function, which is now known for all patterns of length $\le 50$. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-$4$ pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of $1324$-avoiding permutations of length $n$ behaves as \[ B\cdot μ^n \cdot μ_1^{\sqrt{n}} \cdot n^g. \] We estimate $μ=11.600 \pm 0.003$, …

再探错位排列：统计量与模式回避

Desarrangements revisited: statistics and pattern avoidance

本文重新研究首次上升位置为偶数的错位排列，给出按多种排列统计量计数的生成函数公式，并完整枚举回避长度3模式的错位排列，为 Catalan、Fine、Jacobsthal 和 Fibonacci 数提供了新解释。

The stable regularity lemma revisited

We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability.