The distribution of crossings of chords joining pairs of 2𝑛 points on a circle

Riordan, John

doi:10.1090/S0025-5718-1975-0366686-9

The distribution of crossings of chords joining pairs of $2n$ points on a circle
HTML articles powered by AMS MathViewer

by John Riordan PDF

Math. Comp. 29 (1975), 215-222 Request permission

Abstract:

This paper, in the first place, calls attention to an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard. Its main weight, however, is on properties of the several kinds of number sequences appearing.

References

Mém. Acad. Roy. Belgique Coll.

Phys. Rev.

C. L. Mallows and John Riordan, The inversion enumerator for labeled trees, Bull. Amer. Math. Soc. 74 (1968), 92–94. MR 218254, DOI 10.1090/S0002-9904-1968-11888-9
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
John Riordan, Ballots and trees, J. Combinatorial Theory 6 (1969), 408–411. MR 234843, DOI 10.1016/S0021-9800(69)80039-6
Jacques Touchard, Contribution à l’étude du problème des timbres poste, Canad. J. Math. 2 (1950), 385–398 (French). MR 37815, DOI 10.4153/cjm-1950-035-6
Jacques Touchard, Sur un problème de configurations, C. R. Acad. Sci. Paris 230 (1950), 1997–1998 (French). MR 36006
Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math. 4 (1952), 2–25 (French). MR 46325, DOI 10.4153/cjm-1952-001-8
A. M. Yaglom and I. M. Yaglom, Challenging mathematical problems with elementary solutions. Vol. I: Combinatorial analysis and probability theory, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. Translated by James McCawley, Jr.; revised and edited by Basil Gordon. MR 0164902