One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) = | E(P) |. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).

This paper considers the problem of listing all linear extensions of a partial order so that successive extensions differ by the transposition of a single pair of elements. A necessary condition is given for the case when the partial order is a forest. A necessary and sufficient condition is given for the case where the partial order consists of disjoint chains. Some open problems are mentioned. 1 Introduction Many combinatorial objects can be represented by permutations subject to various restrictions. The set of linear extensions of a poset can be viewed as a set of permutations of the elements of the poset. If the Hasse diagram of the poset consists of two disjoint chains, then the linear extension permutations correspond to combinations. If the poset consists of disjoint chains, then the linear extension permutations correspond to multiset permutations. The extensions of the poset that is the product of a 2-element chain with an n-element chain correspond to "ballot sequences" of ...

Let P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P . Call P sign-balanced if the number of linear extensions of P of positive sign is the same as the number of linear extensions of P of negative sign. In this paper we determine when the posets in a particular class are sign-balanced. When posets in this class are not sign-balanced, we determine the difference between the number of positive linear extensions and the number of negative linear extensions. One special case of this class is the product of an m-chain with an n-chain, m and n both ? 1. In this case, we show P is sign-balanced if and only if m = n mod 2.

A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all n-bit necklaces with d ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the last and the first in the list, differ only by the transposition of two bits. The total time required is O(nN (n; d)), where N (n; d) denotes the number of n-bit binary necklaces with d ones. This is the first algorithm for generating necklaces of fixed density which is known to achieve this time bound. 1 Introduction In a combinatorial family, a Gray code is an exhaustive listing of the objects in the family so that successive objects differ only in a small way [Wil]. The classic example is the binary reflected Gray code [Gra], which is a list of all n-bit binary strings in which each string differs from its successor in exactly one bit. By applying the binary Gray code, a variety of problems...

Let B be a basis of transpositions for S n and let Cay(B : S n ) be the Cayley graph of S n with respect to B. It was shown by Kompel'makher and Liskovets that Cay(B : S n ) is hamiltonian. We extend this result as follows. Note that every transposition b in B induces a perfect matching M b in Cay(B : S n ). We show here when n ? 4 that for any b 2 B, there is a Hamilton cycle in Cay(B : S n ) which includes every edge of M b . That is, for n ? 4, for any basis B of transpositions of S n , and for any b 2 B, it is possible to generate all permutations of 1; 2; : : : ; n by transpositions in B so that every other transposition is b.

The graph of linear extensions G(P ) of a poset P has as vertices the linear extensions of P , and two vertices are adjacent if they differ only by an adjacent transposition. This graph has so far been investigated mainly with respect to Hamilton paths and intrinsic geodesic convexity. Especially the work on the latter topic showed how the graph-theoretic properties of G(P ) reflect the order-theoretic structure of P . The aim of this paper is to study the graph G(P ) with respect to topics from classical graph theory, e.g., connectivity, cycle space, isometric embeddings, and nd order-theoretic interpretations of these notions. The main theorems in this paper are the equality of the connectivity of G(P ) and the jump number of P , the existence of a certain generating system for the cycle space and a relationship of P and its subposets obtained via an embedding of the graph of linear extensions.

Combinatorial objects can be represented by strings, such as 21534 for the permutation (1 2) (3 5 4), or 110100 for the binary tree corresponding to the balanced parentheses (()()). Given a string s = s1s2⋯sn, the right-shift operation ��→ shift(s, i, j) replaces the substring sisi+1⋯sj by si+1⋯sjsi. In other words, si is right-shifted into position j by applying the permutation (j j −1 ⋯ i) to the indices of s. Right-shifts include prefix-shifts (i = 1) and adjacent-transpositions (j = i + 1). A fixed-content language is a set of strings that contain the same multiset of symbols. Given a fixedcontent language, a shift Gray code is a list of its strings where consecutive strings differ by a shift. This thesis asks if shift Gray codes exist for a variety of combinatorial objects. This abstract question leads to a number of practical answers. The first prefix-shift Gray code for multiset permutations is discovered, and it provides the first algorithm for generating multiset permutations in O(1)-time while

We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids. For antimatroid (E; F), let J(F) denote the graph whose vertices are the sets of F , where two vertices are adjacent if the corresponding sets differ by one element. Define J(F ; k) to be the subgraph of J(F) 2 induced by the sets in F with exactly k elements. Both graphs J(F) and J(F ; k) are connected, and the former is bipartite. We show that there is a Hamiltonian cycle in J(F) \Theta K 2 . As a consequence, the ideals of any poset P may be listed in such a way that successive ideals differ by at most two elements. We also show that J(F ; k) has a Hamilton path if (E; F) is the poset antimatroid of a series-parallel poset. Similarly, we show that G(L) \Theta K 2 is Hamiltonian, where G(L) is the "basic word graph" of a language antimatroid (E; L). This result was known previously for poset antimatroids. K...

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes Pδ(n,k) and D(n,k) of partitions of n into parts of size at most k in which, for Pδ(n,k), the parts are congruent to one modulo δ and, for D(n,k), the parts are distinct. The change required between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n,k), δ = 1.

Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L1, L2 is the number of pairs of elements of P appearing in different orders in L1 and L2. A diametral pair maximizes the distance among all pairs of linear extensions of P. Felsner and Reuter defined the linear extension diameter of P as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NPcomplete in general, but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: We call a poset diametrally reversing if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing. 1