Name
Abstract | ||
|---|---|---|
A non-crossing pairing on a bit string is a matching of 1s and 0s in the string with the property that the pairing diagram has no crossings. For an arbitrary bit-string w=1^p^"^10^q^"^1...1^p^"^r0^q^"^r, let @f(w) be the number of such pairings. This enumeration problem arises when calculating moments in the theory of random matrices and free probability, and we are interested in determining useful formulas and asymptotic estimates for @f(w). Our main results include explicit formulas in the ''symmetric'' case where each p"i=q"i, as well as upper and lower bounds for @f(w) that are uniform across all words of fixed length and fixed r. In addition, we offer more refined conjectural expressions for the upper bounds. Our proofs follow from the construction of combinatorial mappings from the set of non-crossing pairings into certain generalized ''Catalan'' structures that include labeled trees and lattice paths. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1016/j.jcta.2010.07.002 | J. Comb. Theory, Ser. A |
Keywords | Field | DocType |
arbitrary bit-string w,fixed r,non-crossing pairings,pairing diagram,bit string,combinatorial mapping,fixed length,fuss–catalan numbers,bijective combinatorics,free probability and random matrices,asymptotic estimate,upper bound,non-crossing pairing,catalan number,random matrices,free probability,economics | Discrete mathematics,Combinatorics,Lattice (order),Upper and lower bounds,Diagram,Pairing,Mathematical proof,Bit array,Mathematics,Free probability,Random matrix | Journal |
Volume | Issue | ISSN |
118 | 1 | Journal of Combinatorial Theory, Series A |
Citations | PageRank | References |
1 | 0.48 | 1 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Todd Kemp | 1 | 1 | 0.48 |
| Karl Mahlburg | 2 | 13 | 5.84 |
| Amarpreet Rattan | 3 | 1 | 0.48 |
| Clifford Smyth | 4 | 24 | 6.91 |