Riffle shuffles,cycles, and descents

We derive closed form expressions and limiting formulae for a variety of functions of a permutation resulting from repeated riffle shuffles. The results allow new formulae and approximations for the number of permutations inS _n with given cycle type and number of descents. The theorems are derived from a bijection discovered by Gessel. A self-contained proof of Gessel's result is given.     

A p, q-Analogue of the Generalized Derangement Numbers

In this paper, we study the numbers D _n,k which are defined as the number of permutations σ of the symmetric group S _n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D _n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D _n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D _n,1, denoted by D _n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D _n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D _n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2. Partially supported by NSF grant DMS 0400507.     

Dyck paths and restricted permutations

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and S_n(321) and S_n(231), respectively. We also discuss permutations in S_n avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete continuity between the Motzkin and the Catalan numbers.     

Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

Pattern Avoidance in Multiset Permutations: Bijective Proof

It is well-known that the number of permutations of n letters that avoid a pattern τ of 3 letters is independent of τ. In this note we provide bijective proof that the same result holds for permutations of a multiset. Received August 16, 2006     

Restricted 123-avoiding Baxter permutations and the Padovan numbers

Baxter studied a particular class of permutations by considering fixed points of the composite of commuting functions. This class is called Baxter permutations. In this paper we investigate the number of 123-avoiding Baxter permutations of length n that also avoid (or contain a prescribed number of occurrences of) another certain pattern of length k. In several interesting cases the generating function depends only on k and is expressed via the generating function for the Padovan numbers.     

Restricted Symmetric Permutations

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D ₈, the symmetry group of a square. Three other nontrivial subgroups of D ₈ also have invariant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S ₃. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations. Received September 14, 2006     

Transitivity And Connectivity Of Permutations

It was observed for years, in particular in quantum physics, that the number of connected permutations of [0; n] (also called indecomposable permutations), i. e. those such that for any i < n there exists j > i with (j) < i, equals the number of pointed hypermaps of size n, i. e. the number of transitive pairs (, ) of permutations of a set of cardinality n with a distinguished element.The paper establishes a natural bijection between the two families. An encoding of maps follows. To Chantal, that we may stay connected beyond the simple line of time.     

Counting involutory, unimodal, and alternating signed permutations

In this work we count the number of involutory, unimodal, and alternating elements of the group of signed permutations B_n, and the group of even-signed permutations D_n. Recurrence relations, generating functions, and explicit formulas of the enumerating sequences are given.     

Counting finite posets and topologies

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n11 and all k, and then the numbers of all topologies on n14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 10²³.     

Ann-dimensional search problem with restricted questions

The problem is the following: How many questions are necessary in the worst case to determine whether a pointX in then-dimensional Euclidean spaceR ⁿ belongs to then-dimensional unit cubeQ ⁿ, where we are allowed to ask which halfspaces of (n−1)-dimensional hyperplanes contain the pointX? It is known that ⌌3n/2⌍ questions are sufficient. We prove here thatcn questions are necessary, wherec≈1.2938 is the solution of the equationx log₂ x−(x−1) log₂ (x−1)=1.     

Forest-Like Permutations

Given a permutation , construct a graph G _π on the vertex set {1, 2,..., n} by joining i to j if (i) i < j and π(i) < π(j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π(j). We say that π is forest-like if G _π is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, these show that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which G _π is a tree, or a path, and recover the known generating function of smooth permutations. Received March 27, 2006     

Van der waerden and ramsey type games

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy. By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK _N ^k , and the first player wins if he has selected allk-tuples of ann-set. LetR _k*(n) be the least integerN so that the first player has a winning strategy. We prove (W* (n))¹/n → 2,R ₂*(n)<(2+ε) ⁿ andR _k ^* n<2 ^nk / ^k! fork ≧3.     

Codes And Xor Graph Products

What is the maximum possible number, f₃(n), of vectors of length n over {0,1,2} such that the Hamming distance between every two is even? What is the maximum possible number, g³(n), of vectors in {0,1,2}ⁿ such that the Hamming distance between every two is odd? We investigate these questions, and more general ones, by studying Xor powers of graphs, focusing on their independence number and clique number, and by introducing two new parameters of a graph G. Both parameters denote limits of series of either clique numbers or independence numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of the series grows exponentially as the power tends to infinity, while the other grows linearly. As a special case, it follows that f₃(n) = Θ(2ⁿ) whereas g₃(n)=Θ(n). * Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. † Research partially supported by a Charles Clore Foundation Fellowship.     

VC-Dimension of Sets of Permutations

VC-dimension of a set of permutations to be the maximal k such that there exist distinct that appear in A in all possible linear orders, that is, every linear order of is equivalent to the standard order of for at least one permutation . In other words, the VC-dimension of A is the maximal k such that for some the restriction of A to contains all possible linear orders. This is analogous to the VC-dimension of a set of strings. Our main result is that there exists a universal constant C such that any set of permutations with VC-dimension 2 is of size . This is analogous to Sauer's lemma for the case of VC-dimension 2. One corollary of our main result is that any acyclic set of linear orders of is of size , (a set A of linear orders on is called acyclic if no 3 elements appear in A in all 3 orders (i,j,k), (k,i,j) and (j,k,i)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of n alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive. Received August 6, 1998     

Three enumeration formulas of standard Young tableaux of truncated shapes

In this paper we consider the enumeration of three kinds of standard Young tableaux (SYT) of truncated shapes by use of the method of multiple integrals. A product formula for the number of truncated shapes of the form (n^m, n ? r)\δ_k–1 is given, which implies that the number of SYT of truncated shape (n², 1)\(1) is the number of level steps in all 2-Motzkin paths. The number of SYT with three rows truncated by some boxes ((n + k)³)\(k) is discussed. Furthermore, the integral representation of the number of SYT of truncated shape (n^m)\(3, 2) is derived, which implies a simple formula of the number of SYT of truncated shape (nⁿ)\(3, 2).     

Maximizing the descent statistic

For a subsetS, let the descent statistic (S) be the number of permutations that have descent setS. We study inequalities between the descent statistics of subsets. Each subset (and its complement) is encoded by a list containing the lengths of the runs. We define two preorders that compare different lists based on the descent statistic. Using these preorders, we obtain a complete order on lists of the form (k ⁱ ,P,k ^n–i , whereP is a palindrome, whose first entry is larger thank. We prove a conjecture due to Gessel, which determines the list that maximizes the descent statistic, among lists of a given size and given length. We also have a generalization of the boustrophedon transform of Millar, Sloane and Young.     

Some asymptotic results useful in enumeration problems

Suppose one is given two related generating functionsa(x) = a _n x ⁿ andb(x) = b _n x ⁿ , often it is of interest to determine the limiting behaviour of the quantitiesa _n /b _n We survey some earlier results of this nature and give some new ones     

Dense Packing of Patterns in a Permutation

We study the length L _k of the shortest permutation containing all patterns of length k. We establish the bounds e ⁻² k ² < L _k ≤ (2/3 + o(1))k ². We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k ² containing almost all patterns of length k. Received January 2, 2007     

Some asymptotic results useful in enumeration problems

Suppose one is given two related generating functionsa(x) = a _n x ⁿ andb(x) = b _n x ⁿ , often it is of interest to determine the limiting behaviour of the quantitiesa _n /b _n We survey some earlier results of this nature and give some new ones