Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl

Abstract. Another bijective proof of Stanley’s hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author’s previous bijective proof (“An involution principle-free bijective proof of Stanley’s hook-content formula”, Discrete Math. Theoret. Computer Science, to appear) and the Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1 (1997), 53–67) for the hook formula for standard Young tableaux of a given shape. This new algorithm can also be used as an algorithm for the random generation of tableaux of a given shape with bounded entries. An appropriate deformation of this algorithm gives an algorithm for the random generation of plane partitions inside a given box. 1. Introduction. There

this article is to give a bijective proof for Stanley's hook-content formula [15, Theorem 15.3] for a certain plane partition generating function. In order to be able to state the formula we have to recall some basic notions from partition theory. A partition is a sequence = ( 1 ; 2 ; : : : ; r ) with 1 2 r > 0, for some r. The Ferrers diagram of is an array of cells with r leftjustified rows and i cells in row i. Figure 1.a shows the Ferrers diagram corresponding to (4; 3; 3; 1). The conjugate of is the partition ( 1 ; : : : ; 1 ) where j is the length of the j-th column in the Ferrers diagram of . We label the cell in the i-th row and j-th column of (the Ferrers diagram of) by the pair (i; j). Also, if we write 2 we mean ` is a cell of '. The hook length h of a cell = (i; j) of is ( i j) + ( j i) + 1, the number of cells in the hook of , which is the set of cells that are either in the same row as and to the right of , or in the same column as and below , included. The content c of a cell = (i; j) of is j i

ABSTRACT. — We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments. Summary §1. Introduction. Selected hook formulas. Conjecture §2. Classical hook length formulas for partitions. §3. Hook length expansion algorithm and HookExp. §4. The exponent principle. §5. Hook length formulas for partitions.

Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect all the currently existing references to the papers directly related to the subject. Moreover, a number of basic approaches to study the pattern problems are discussed.

We give alternative proofs of determinantal formulas for certain up-down tableaux and down-up tableaux generating functions, which were first given in another paper by the author. These new proofs are based on interpreting up-down and down-up tableaux as certain families of nonintersecting lattice paths with three types of steps.