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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer… (More)

- Victor G Kac, Weiqiang Wang, Catherine H Yan, V G Kac, W Wang, C H Yan
- 1998

We show that there are precisely two, up to conjugation, anti-involutions σ ± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D ± of the Lie subalgebra of this algebra fixed by −σ ± , and find the unitary ones. We… (More)

- Dimitrije Kosti´c, Catherine H Yan
- 2007

A parking function of length n is a sequence (b 1 , b 2 ,. .. , b n) of nonnegative integers for which there is a permutation π ∈ S n so that 0 ≤ b π(i) < i for all i. A well-known result about parking functions is that the polynomial P n (q), which enumerates the complements of parking functions by the sum of their terms, is the generating function for the… (More)

- Catherine H Yan
- 2007

A generalized x-parking function associated to x = (a, b, b,. .. , b) ∈ AE n is a sequence (a 1 , a 2 ,. .. , a n) of positive integers whose non-decreasing rearrangement b 1 ≤ b 2 ≤ · · · ≤ b n satisfies b i ≤ a + (i − 1)b. The set of x-parking functions is equinumerate with the set of sequences of rooted b-forests on [n]. We construct a bijection between… (More)

A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz, et.al. [5] that for any matrix A, lindisc(A) ≤ herdisc(A). (1) When A is the incidence matrix of a… (More)

In his paper (1942), Ore found necessary and sufficient conditions under which the modular and distributive laws hold in the lattice of equivalence relations on a set S. In the present paper, we consider commuting equivalence relations. It has been proved by J6nsson (1953) that the modular law holds in the lattice of commuting equivalence relations. We give… (More)

A non-crossing pairing on a binary string pairs ones and zeroes such that the arcs representing the pairings are non-crossing. A binary string is well-balanced if it is of the form 1 a 1 0 a 1 1 a 2 0 a 2. .. 1 ar 0 ar. In this paper we establish connections between non-crossing pairings of well-balanced binary strings and various lattice paths in plane. We… (More)