On permutations with bounded drop size

Descent polynomials for permutations with bounded drop size

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1, 2, . . . , n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients. Résumé. Motivés par les “suites de jonglerie” et le tri à bull...

Geometric permutations of balls with bounded size disparity

We study combinatorial bounds for geometric permutations of balls with bounded size disparity in d-space. Our main contribution is the following theorem: given a set S of n disjoint balls in R , if n is sufficiently large and the radius ratio between the largest and smallest balls of S is γ , then the maximum number of geometric permutations of S is O(γ logγ ). When d = 2, we are able to prove ...

Geometri Permutations of Balls with Bounded Size Disparity

Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of n-permutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(n logα(n)) and for every t ≥ 1, we have r2...

Polynomial permutations on bounded commutative directoids with an antitone involution

Main results of Dorninger and Länger (J Pure Appl Math 40:441–449, 2007) concerning polynomial permutations on bounded lattices with an antitone involution are generalized to the case of bounded commutative directoids.