We present a number of conjectures involving rook polynomials having only real zeros. Many of these generalize a previous conjecture of the author, K. Ono, and D. G. Wagner, namely that if A is a real n n matrix which is weakly increasing down columns, then the permanent of zA + Jn has only real zeros. In some cases these conjectures are motivated by factorization theorems for Ferrers boards. S...

Abstract. We investigate the combinatorial properties of the traces of the n-th Hecke operators on the spaces of weight 2k cusp forms of level N . We establish examples in which these traces are expressed in terms of classical objects in enumerative combinatorics (e.g. tilings and Motzkin paths). We establish in general that Hecke traces are explicit rational linear combinations of values of Ge...

We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production rules whose rule operators commute. In this situ...

The notion of (3+1)-avoidance appears in many places in enumerative combinatorics, but the natural goal of enumerating all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets. Our proof consists of a number of structural theorems followed by some generating function magic. Résumé. L’idée de l’évitement de (3+1) apparaı̂t dans beaucoup d’endroits dans le c...

Anne Pieper currently teaches high school geometry at The Westminster Schools in Atlanta, Georgia. She recently graduated fromThe University of Georgia with an M.A. in Mathematics Education. Current trends in the revision of both mathematics curricula and traditional teaching strategies include the incorporation of discrete mathematics into the secondary school curriculum (Dossey, 1990). Seriou...

We address a well-known problem in combinatorics involving the identification of counterfeit coins with a systematic approach. The methodology can be applied to cases where the total number of coins is exceedingly large such that brute force or enumerative comparisons become impractical. Based on the principle behind this approach, the minimum effort necessary for identification of the counterf...

We investigate the combinatorial properties of the traces of the nth Hecke operators on the spaces of weight 2k cusp forms of level N. We establish examples in which these traces are expressed in terms of classical objects in enumerative combinatorics (e.g., tilings and Motzkin paths). We establish in general that Hecke traces are explicit rational linear combinations of values of Gegenbauer (a...

The notion of (3+1)-avoidance appears in many places in enumerative combinatorics, but the natural goal of enumerating all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets. Our proof consists of a number of structural theorems followed by some generating function magic. Résumé. L’idée de l’évitement de (3+1) apparaı̂t dans beaucoup d’endroits dans le c...

In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences’ terms, our problem can be really difficult, which calls for novel approaches to work it out for any given pair (m,n). Complete or partial solutions for m = 2, 3 with smaller n’s are listed. Moreover, we have proved the necessa...

We present a theorem-proving experiment performed with a computer algebra system. It proves a conjecture about the general pattern of the generating functions counting rooted maps of given genus. These functions are characterized by a complex non-linear differential system between generating functions of multi-rooted maps. Establishing a pattern for these functions requires a sophisticated indu...