Enumeration Of Partitions Research Articles

Enumeration of partitions with prescribed successive rank parity blocks

Successive ranks of a partition, which were introduced by Atkin, are the difference of the lengths of the i-th row and the i-th column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parity blocks. Motivated by his work, we investigate partitions with prescribed successive rank parity blocks. The main result of this paper is the generating function of partitions with exactly d successive ranks and m parity blocks.

Some Restricted Partition Functions

In 1742, Euler found the generating function for P(n). Hardy said Ramanujan was the first, and upto now the only, Mathematician to discover any such properties of P(n). In 1952, Macmahon established a table of P(n) for the first 200 values of n. This paper showed how to find the number of partition of n by using Macmahon’s table. In 1742, Euler also stated the series in the enumeration of partitions. This Paper showed how to generate the Euler’s use of series in the enumeration of partitions. In 1952, Macmahon also quoted the self-conjuate partitions of n. In this Paper, Macmahon’s self-conjugate partitions are explained with the help of array of dots. This paper showed how to prove the Euler’s Theorems with the help of Euler’s device of the introduction of a second parameter z, and showed how to prove the Theorem 3 with the help of Euler’s generating function for P(n), and also showed how to prove the Theorem 4 with the help of certain conditions of P(n).

Note on enumeration of partitions contained in a given shape

Carlitz, Handa, and Mohanty proved determinantal formulas for counting partitions contained in a fixed bounding shape by area. Gessel and Viennot introduced a combinatorial method for proving such formulas by interpreting the determinants as counting suitable configurations of signed lattice paths. This note describes an alternative combinatorial approach that uses sign-reversing involutions to prove matrix inversion results. Combining these results with the classical adjoint formula for the inverse of a matrix, we obtain a new derivation of the Handa–Mohanty determinantal formula.

Block-connected set partitions

This paper introduces two statistics on set partitions, namely connector and circular connector. If B 1 / … / B k is a partition of { 1 , … , n } with k > 1 blocks, then a connector is an ordered pair ( c , c + 1 ) satisfying c ∈ B i , c + 1 ∈ B i + 1 , i = 1 , … , n − 1 . A circular connector is a connector when the blocks of a partition are arranged on a circle. We concentrate on the enumeration of partitions according to the two statistics, and certain variations thereof. Our results include several nice generating functions and explicit formulas. We also establish connections between connected partitions and words over a finite alphabet, and random walks on a square lattice.

Configurational entropy of codimension-one tilings and directed membranes

The calculation of random tiling configurational entropy amounts to an enumeration of partitions. A geometrical description of the configuration space is given in terms of integral points in a high-dimensional space, and the entropy is deduced from the integral volume of a convex polytope. In some cases the latter volume can be expressed in a compact multiplicative formula, and in all cases in terms of binomial series, the origin of which is given a geometrical meaning. Our results mainly concern codimension-one tilings, but can also be extended to higher codimension tilings. We also discuss the link between freeboundary-and fixed-boundary-condition problems.

Enumeration of Partitions with Hooklengths

Partitions of an integer have found extensive application in combinatorics [1, 14, 17], group representation theory [6, 13, 15], and the theory of algorithms [10, 12]. The component parts of a partition can be arranged linearly, in the plane or even associated with the elements of an arbitrary partially ordered set. One of the important properties of linear partitions is that their generating function can be written

A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or $6 - j$ Symbols

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.

A new family of partition identities

1. George E. Andrews and Richard Askey, Enumeration of Partitions, Proceedings ofthe NATO Advanced Study Institute Higher Combinatorics, Berlin, Sept. 1-10, 1976,ed. by M. Aigner, publ. by Reidel, Dordrecht.2. G. H. Hardy and E. M. Wright, An Introduction on the Theory of Numbers, fourthedition, Oxford.Received December 28, 1976.

Series Connected with the Enumeration of Partitions

Proceedings of the London Mathematical SocietyVolume s2-1, Issue 1 p. 63-88 Papers Series Connected with the Enumeration of Partitions REV. F. H. Jackson, REV. F. H. JacksonSearch for more papers by this author REV. F. H. Jackson, REV. F. H. JacksonSearch for more papers by this author First published: 1904 https://doi.org/10.1112/plms/s2-1.1.63Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-1, Issue11904Pages 63-88 RelatedInformation