Related papers: Signed Enumeration of Upper-Right Corners in Path …
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This…
For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact…
We show that there is the same number of (n,l)-alternating sign trapezoids as there is of column strict shifted plane partitions of class l-1 with at most n parts in the top row, thereby proving a result that was conjectured independently…
We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we…
We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of $\sqrt n$ and directional statistics for a shifted lattice. We show that the randomly rotated, and…
Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these polynomials which refine…
Stirling numbers, which count partitions of a set and permutations in the symmetric group, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the…
Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Bj{\"o}rner. In this paper, we give an axiomatic treatment of either concept and…
We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a "mating-of-trees" type bijection. This class includes the…
A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed…
This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for…
We study the problem of counting real simple rational functions $\varphi$ with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus $0$). We introduce a signed count of such functions that is…
Signed shifts are generalizations of the shift map in which, interpreted as a map from the unit interval to itself sending x to the fractional part of Nx, some slopes are allowed to be negative. Permutations realized by the relative order…
We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the…
We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{\'e}lou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed.…
Juggling patterns can be mathematically modeled as closed walks within directed state graphs. In this paper, we present a unified framework of unbounded juggling patterns and its variations (including multiplex, colored, and passing)…
We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to…
Recently Albert and Bousquet-M\'elou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We…
Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained.…
We give another proof for the (-1)-enumeration of self-complementary plane partitions with at least one odd side-length by specializing a certain Schur function identity. The proof is analogous to Stanley's proof for the ordinary…