By Daniel A. Marcus

This publication teaches the artwork of enumeration, or counting, by way of best the reader via a sequence of rigorously selected difficulties which are prepared strategically to introduce innovations in a logical order and in a provocative manner. it's prepared in 8 sections, the 1st 4 of which hide the elemental combinatorial entities of strings, mixtures, distributions, and walls. The final 4 conceal the distinct counting tools of inclusion and exclusion, recurrence family members, producing services, and the equipment of Pуlya and Redfield that may be characterised as "counting modulo symmetry. the original layout combines positive factors of a standard textbook with these of an issue booklet. the subject material is gifted via a chain of roughly 250 difficulties, with connecting textual content the place applicable, and is supplemented by means of nearly 2 hundred extra difficulties for homework assignments. Many purposes to likelihood are incorporated during the ebook. whereas meant essentially to be used because the textual content for a college-level direction taken via arithmetic, laptop technological know-how, and engineering scholars, the ebook is acceptable to boot for a basic schooling direction at a very good liberal arts collage, or for self examine.

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Now that we have pentagonal numbers, we can define m-gonal numbers for any m ~ 3 in a similar manner. The two most common types of 40 12. Classical Partition Function and Euler's Product Formula polygonal numbers are the triangular numbers, ~ _ n(n+ 1) 2 n - ' and , of course, the square numbers , On = n 2 • In general , we may deduce geometrically the formula for the nth m-gonal number: mn _ (2)~ - m - + n -_ n(mn-2n-m+4) 2 . n -l Identities for triangular and square numbers similar to Euler 's product formula can also be derived from Jacobi's triple product identity.

1. ~~ . 6) Definition. 9) 9. Two Euler's Identities and Two q-Exponential Functions 31 Analogously, we can define another q-exponential function using El. Definition. Another q-analogue of the classical exponential function is EX = q ~ qj(j-1 )/2 x j [j]! J J=O = (1 + (1 _ q)x)oo . 10) q Let us study some properties of the two q-exponential functions. The classical exponential function is unchanged under differentiation. Its two q-analogues have similar behavior. J -[j]! J -[j]! J [ . -I]! ' J=O J=l J=l J J=O J and , t qj(j-1)/2 D~~j = [J] .

D Like the Pascal rule, many identities involving binomial coefficients have their q-analogues. Imagine that we have m + n balls, and they are placed into two groups, one with m and one with n of them. Each way of choosing k balls from all m + n of them corresponds in a one-to-one manner to a way of choosing j balls from the group with m balls and choosing k - j balls from the group of n balls , with j running from 0 to k. ). 3) Example. 1. Let V = IF;:+n and let Vm C V be a fixed subspace with dim Vm = m .