Lily Yen

Set Partitions with No m-Nesting

A partition of \(\{1,\ldots,n\}\) has an m-nesting if it contains at least m disjoint blocks, and a subset of 2m points \(i_{1} < i_{2} <\ldots < i_{m} < j_{m} < j_{m-1} <\ldots < j_{1}\), such that i l and j l are in the same block for all 1 ≤ l ≤ m, but no other pairs are in the same block. In this note, we use generating trees to construct the class of partitions with no m-nesting, determine functional equations satisfied by the associated generating functions, and generate enumerative data for m ≥ 4.

On the length of integers in telescopers for proper hypergeometric terms

We show that the number of digits in the integers of a creative telescoping relation of expected minimal order for a bivariate proper hypergeometric term has essentially cubic growth with the problem size. For telescopers of higher order but lower degree we obtain a quintic bound. Experiments suggest that these bounds are tight. As applications of our results, we give an improved bound on the maximal possible integer root of the leading coefficient of a telescoper, and the first discussion of the bit complexity of creative telescoping.

A Note on Multiset Permutations

The author studies permutations of the multiset

A Combinatorial Proof for Stockhausen's Problem

\{1,1,2,2,\ldots,m,m,m+1,m+2\ldots,n\}

A Generating Tree Approach to k-Nonnesting Partitions and Permutations

such that

Crossings and Nestings for Arc-Coloured Permutations

1,2,\ldotsn

Constructing Skolem sequences via generating trees

occurs as a not-necessarily consecutive subsequence. From the theory of symmetric functions, the generating function for the number of these permutations is known [Goulden and Jackson, Combinatorial Enumeration, John Wiley, New York, 1983, p. 73]. It is used to obtain a recurrence relation and then to give a purely combinatorial proof of the recurrence.

A Bijection for Crossings and Nestings

We consider problems in the enumeration of sequences suggested by the problem of determining the number of ways of performing a piano composition (Klavierstuck XI) by Karlheinz Stockhausen. An explicit formula and a combinatorial proof for the general problem are given.