Cayley's Theorem Research Articles

A combinatorial proof of Cayley's theorem on Pfaffians

Cayley's theorem, published in 1847, asserts that any skew-symmetric determinant of even order is equal to the square of its Pfaffian. A proof of this result, depending on the combinatorial structures of determinants and Pfaffians, is presented here. It emerges that the theorem holds, essentially because the representation of a permutation of [1, 2,…, 2 m] as a product of disjoint cycles of odd parity relates it biuniquely to two independent partitions of the same set into pairs, while a permutation whose cyclic representation contains cycles of even parity corresponds to a term of the determinant which is eliminated by skew-symmetry.

An identity of the jacobi type for pfaffians

The paper refers to the well-known identity, published by Jacobi in 1833, relating each minor of a determinant to the complementary minor of the adjugate determinant. An exactly analogous identity is proved for Pfaffians, making use of Cayley's theorem of 1847, relating a skew-symmetric determinant to its Pfaffian.

The codifying of tree-structure

1. The mathematician's tree is a collection of pairs of elements, conveniently called points. A pair of points is a link, and a collection of links is a tree if any two of the points from which the links are formed can be connected by a chain, and if there are no closed chains. A tree is determined if there is provided a complete schedule of the links composing the tree. Conversely, no description of a tree can be adequate unless a complete schedule is recoverable from the description. An arbitrary schedule of links does not usually describe a tree, and the object of this note is to give canonical schedules for the most general finite tree, and to specify these schedules as economically as possible. The fundamental result established is that the structure of a tree with n points can be encoded in a row of n − 2 symbols each of which may stand for any one of the points. Cayley's theorem, that the number of different trees with n given points is nn−2, is an immediate corollary.

On the Expansion of Devertebrated Three Dimensional Determinants and the Extension of Cayley's Expansion Theorem

On the Expansion of Devertebrated Three Dimensional Determinants and the Extension of Cayley's Expansion Theorem

On the Expansion of Devertebrated Three Dimensional Determinants and the Extension of Cayley's Expansion Theorem

On the Expansion of Devertebrated Three Dimensional Determinants and the Extension of Cayley's Expansion Theorem

Scientific Serials

American Journal of Mathematics, vol. ix. No. 2, January.—The number opens with a continuation of Mr. Greenhill's memoir, wave-motion in hydrodynamics, in which is discussed wave-motion in the following cases: § 21, across a channel with sides sloping at any angle; § 22, against a uniformly-sloping shore; § 24, in a cone; § 25, in a cylinder; and § 23 contains an algebraical solution of waves against a shore.—Prof. Sylvester's lectures on the theory of reciprocants give notes of lectures xvii. to xxiv., with an extract from a letter of M. Halphen in which the existence of invariants in general is established a priori; this is given as introductory to the theory of differential invariants.—A memoir in the theory of numbers, by A. S. Hathaway, contains an historical introduction of interest. The second part considers fundamental principles and definitions, then a problem and the consequences of its solution, and then turns the question of ideal solution of the problem into the question of the establishment of a given theory of ideals; the demonstrations are left for the reader to supply. The third part is occupied with a rigorous establishment of the theory of ideals indicated in the second part.—The next paper, on a theorem respecting the singularities of curves of multiple curvature, by H. B. Fine, is a generalisation of a portion of a previous paper (vol. viii. No. 2) by the same writer.—The number closes with two short notes—one on pencils of conies, by H. D. Thompson (let the eight points in which a conic intersects a quartic be divided into two groups of four, and a conic be passed through each group: the two residual—four-point—groups lie on a conic; an exceptional case in Cayley's theorem, which had been overlooked by the author, is mentioned and references given to where it is discussed); the other consists of observations on the generating functions of the theory of invariants, by Capt. P. A. Macmahon.

3. On Professor Cayley's Theorem regarding a Bordered Skew Determinant

3. On Professor Cayley's Theorem regarding a Bordered Skew Determinant