A Formal Calculus for the Enumerative System of Sequences—I. Combinatorial Theorems

Abstract

The enumeration of sequences plays an important part in combinatorial enumeration since sequences may be used as a means for encoding combinatorial structures. In the main, the enumeration of sequences has been treated by a collection of methods individually applied in particular cases. We present here a uniform constructive method for dealing with a large class of sequence problems. We obtain the generating function as the trace of an expression involving certain incidence matrices, expressions which are obtainable immediately from the combinatorial structure of a problem. The method relies on certain properties of matrices whose rank is equal to one. We demonstrate that these properties may be used to derive the required generating functions explicitly or, in more complex cases, as solutions to systems of linear equations. Results for permutations may be derived from those for sequences, and in certain frequently encountered special cases this operation may be carried out by a straightforward substitution. Applications of the method presented here will be given in Part II.

A Logarithmic Connection for Circular Permutation Enumeration

“…Many other applications of Lemma 3.1 to the enumeration of sequences and permutations are given in [4,7,8,9,10,11]. One further example is included here to demonstrate that H need not be a monomial in A and B, as it was above, but may be a power series in A and B, with additional indeterminates, retaining combinatorial information, as coefficients.…”

“…Only a small subset of the possible applications of our method has been presented here. In fact, there seems to be a circular analogue of all of the sequence enumeration problems considered in [7,8,9,10,11]. These include extension to the situation in which the pattern can involve nonadjacent elements (the case of linear permutations is considered in [10)), as well as consideration of sets of simultaneous circular permutations.…”

“…The purpose of this paper is to present a method for enumerating unrooted circular sequences under certain restrictions and with respect to various characteristics. The enumeration of sequences in .9" has been considered in some depth in [7,8,9,10,11]. We have demonstrated in these papers that the required generating function is one component of a vector of functions which satisfies a linear system with coefficient matrix M, say.…”

A Matrix for Counting Paths in Acyclic Digraphs

“…The following result appears in [3,Lemma 3.12]. Let R and S be n_n matrices such that R+S=J, the all 1's matrix.…”

A combinatorial application of matrix Riccati equations and their q-analogue

“…2>"u = pu (1.3) 0012-365X/81/0000-0000/$02.50 © North-Holland for some fixed /3 and fixed positive integer a, and are hence obtained in closed form. Equation (1.2) was previously obtained using a purely algebraic argument by Jackson and Goulden [7] and Gessel [6] attributes a number of expressions of this type to Stanley [12]. A particular case has been given by Carlitz and Scoville [21…”