A pitch tournament is a resolvable or near resolvablevY 8Y 7 BIBD that satis®es certain criteria in addition to theusual condition that v 0 or 1 mod 8. Here we establish that for the case v 8n the necessary condition forpitch tournaments is suf®cient for all n b 1615, with at most 187 smaller exceptions. This complements our earlier study of the v 8n 1 case, where we established suf®ciency for all n b 224, with at most 28 smaller exceptions. The four missing cases for vY 8Y 7 BIBDs are provided, namely v Pf48Y 56Y 96Y 448g, thereby establishing that the necessary existence conditions are suf®cient without exception. Some constructions for resolvable designs are also provided, reducing the existence question for vY 8Y 7 RBIBDs to 21 possible exceptions.

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It is well known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational˛. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers˛. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r-Fibonacci sequences, and the Pell sequence, to name a few. We show that these sequences satisfy a linear recurrence relation when considered modulo k, even though the sequences themselves do not. We then employ this recurrence relation to determine the generating functions and Binet-like formulas. We end by discussing the convergence of the ratios of the terms of these sequences.

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A 1 ⊕ A 1 , A 2 , C 2 , and G 2 . Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.