Enumeration of derangements with descents in prescribed positions

We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated

Enumeration of derangements with descents in prescribed positions

We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated

The excedances and descents of bi-increasing permutations

Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that these (so-called bi-increasing) permutations are just the 321-avoiding ones. The paper investigates their excedance and descent structure. In particular, we find some nice combinatorial interpretations for the distribution coefficients of the number of excedances and descents, respectively, and their difference analogues over the bi-increasing permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This yields a connection between restricted permutations, parallelogram polyominoes, and lattice paths that reveals the relations between several well-known bijections given for these objects (e.g. by Delest-Viennot, Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an application, we enumerate skew diagrams according to their rank and give a simple combinatorial proof for a result concerning the symmetry of the joint distribution of the number of excedances and inversions, respectively, over the symmetric group.Comment: 36 page

Topics in algorithmic, enumerative and geometric combinatorics

This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S_2xS_{n-2}-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai's 3^d-conjecture for such polytopes. Furthermore, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given. We end by constructing an interesting class of Hansen polytopes having very few faces and yet not being Hanner. The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the packing density for some classes of generalized patterns, including all the three letter patterns. In the fourth paper, we present combinatorial proofs of the enumeration of derangements with descents in prescribed positions. To this end, we consider fixed point lambda-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n. The fifth paper solves a partially ordered generalization of the famous secretary problem. The elements of a finite nonempty partially ordered set are exposed in uniform random order to a selector who, at any given time, can see the relative order of the exposed elements. The selector's task is to choose online a maximal element. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order, thus proving that the linearly ordered set is at least as difficult as any other instance of the problem. To this end, we define a probability measure on the maximal elements of an arbitrary partially ordered set, that may be interesting in its own right