The Euler series transformation and the binomial identities of Ljunggren, Munarini and Simons.
The (Euler)-Catalan numbers.
The expected variation of random bounded integer sequences of finite length.
The extension of a cyclic inequality to the symmetric form.
The factorial representations of integers and the Eratosthenes sieve
The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion.
The Fibonacci number of a grid graph and a new class of integer sequences.
The fifth taxicab number is 48988659276962496.
The finite Heine transformation and conjugate Durfee squares.
The fraction of subspaces of with a specified number of minimal weight vectors is asymptotically Poisson.
The freeness of ideal subarrangements of Weyl arrangements
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our...
The generalized Schröder theory.
The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles
The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...
The generating function of ternary trees and continued fractions.
The half-perimeter generating function of gated and wicketed Ferrers diagrams.
The hyperharmonic numbers and the phratry of the coupon collector. (Les nombres hyperharmoniques et la fratrie du collectionneur de vignettes.)
The identity of three classes of polynomials.
The initial involution patterns of permutations.
The insertion encoding of permutations.
The integer sequence A002620 and upper antagonistic functions.