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A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example, should we represent the subset ${2,5}$ of ${1,2,3,4,5}$ as 25 in a linear string? Is the representation 52 acceptable? Or it it tactically advantageous (and acceptable) to go with ${0,1,0,0,1}$? In this paper, we represent combinatorial objects as graphs, as in cite{bks}, and exhibit the flexibility and power of this representation to produce {it graph universal cycles}, or {it Gucycles}, for $k$-subsets of an $n$-set; permutations (and classes of permutations) of $[n]={1,2,ldots,n}$, and partitions of an $n$-set, thus revisiting the classes first studied in cite{cdg}. Under this graphical scheme, we will represent ${2,5}$ as the subgraph $A$ of $C_5$ with edge set consisting of ${2,3}$ and ${5,1}$, namely the second and fifth edges in $C_5$. Permutations are represented via their permutation graphs, and set partitions through disjoint unions of complete graphs.
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A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian, this baseline result is used as the basis of existence proofs for universal cycles (also known as generalized deBruijn cycles or U-cycles) of several combinatorial objects. We extend the body of known results by presenting new results on the existence of universal cycles of monotone, augmented onto, and Lipschitz functions in addition to universal cycles of certain types of lattice paths and random walks.
It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and passwords.
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Section 4 is an enumeration of point-determining bipartite graphs using this toolset. In Section 5, we extend a result about point-determining graphs to a similar result for point-determining {Phi}-graphs, where {Phi} is a class of graphs with certain properties. Finally, Appendix A is an expository on species computation using the software Sage [9] and Appendix B uses Sage to calculate the cycle index series of point-determining bipartite graphs.
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial proofs, including results of Boccara (1980), Zagier (1995), Stanley (2011), F{e}ray and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new results as well. In particular, for the first time, given a partition of a set, we obtain an explicit formula for the number of pairs of long cycles on the set such that the product of the long cycles does not mix the elements from distinct blocks of the partition and has an independently prescribed number of cycles for each block of elements. As applications, we obtain new explicit formulas concerning factorizations of any even permutation into long cycles and the first nontrivial explicit formula for computing strong separation probabilities solving an open problem of Stanley (2010).
The Bubble-sort graph $BS_n,,ngeqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set ${(1 2), (2 3),ldots, (n-1 n)}$. It is a bipartite graph containing all even cycles of length $ell$, where $4leqslant ellleqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,,ngeqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.
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