Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.

Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

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. This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n) 3) in approximate-size sampling and O(n4/3) in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a × b)-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

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