The 11-element case of Frankl's conjecture.
The active bijection between regions and simplices in supersolvable arrangements of hyperplanes.
The average order of a permutation.
The box parameter for words and permutations
The box parameter for words counts how often two letters w j and w k define a “box” such that all the letters w j+1; ..., w k−1 fall into that box. It is related to the visibility parameter and other parameters on words. Three models are considered: Words over a finite alphabet, permutations, and words with letters following a geometric distribution. A typical result is: The average box parameter for words over an M letter alphabet is asymptotically given by 2n − 2n H M/M, for fixed M and n → ∞.
The calculation of α-sets of representatives
The derangement problem relative to the Mahonian process.
The descent set and connectivity set of a permutation.
The dinner table problem: the rectangular case.
The enumeration of simple permutations.
The Erdős-Heilbronn problem for finite groups
The fifth taxicab number is 48988659276962496.
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 initial involution patterns of permutations.
The insertion encoding of permutations.
The Josephus problem
We give explicit non-recursive formulas to compute the Josephus-numbers and and explicit upper and lower bounds for (where ) which differ by (for the bounds are even better). Furthermore we present a new fast algorithm to calculate which is based upon the mentioned bounds.
The Largest Subdeterminant of a Matrix
The many formulae for the number of Latin rectangles.
The minimum number of monotone subsequences.
The Möbius function of the consecutive pattern poset.
The number of ternary words avoiding abelian cubes grows exponentially.