Download PDF by Martin Aigner: A Course in Enumeration

By Martin Aigner

ISBN-10: 3540390324

ISBN-13: 9783540390329

Combinatorial enumeration is a with ease obtainable topic packed with simply said, yet occasionally tantalizingly tricky difficulties. This ebook leads the reader in a leisurely approach from the fundamental notions to a number of themes, starting from algebra to statistical physics. Its target is to introduce the scholar to a fascinating box, and to be a resource of data for the pro mathematician who desires to research extra in regards to the topic. The ebook is prepared in 3 components: fundamentals, tools, and themes. There are 666 routines, and as a distinct characteristic each bankruptcy ends with a spotlight, discussing a very attractive or well-known result.

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Example text

7) For Am,n (q) we split the paths into two classes, depending on whether for the largest summand, λ1 < m or λ1 = m. In the first case we obtain Am−1,n (q), and in the second (after deleting the top row containing m dots) qm Am,n−1 (q). Hence Am,n (q) = Am−1,n (q) + qm Am,n−1 (q) , and this is precisely recurrence (7). As a corollary we can state the q-binomial theorem generalizing n n (x + 1)n = k=0 k x k . 2. We have n 2 n k n (1 + xq)(1 + xq ) · · · (1 + xq ) = k=0 Proof. k+1 2 q q( )xk . Expanding the left-hand side we obtain n (1 + xq) · · · (1 + xqn ) = bk (q)x k , k=0 where q|λ| .

N} in increasing order, the second line is a unique n-permutation. We call σ = σ (1)σ (2) . . σ (n) the word representation of σ . Another way to describe σ is by its cycle decomposition. For every i, the sequence i, σ (i), σ 2 (i), . . must eventually terminate with, say, σ k (i) = i, and we denote by i, σ (i), σ 2 (i), . . , σ k−1 (i) the cycle containing i. Repeating this for all elements, we arrive at the cycle decomposition σ = σ1 σ2 · · · σt . Example. σ = 12345678 35146827 has word representation σ = 35146827 and cycle form σ = (13)(25687)(4).

Use the previous exercise to prove the “Stirling binomial theorem” pn (x + y) = n k=0 n k pk (x)pn−k (y). 34 Determine the number f (n, k) of sequences a1 a2 . . an of positive integers such that the largest entry is k, and the first occurrence of i appears before the first occurence of i + 1 (1 ≤ i ≤ k − 1). Hint: f (n, k) = Sn,k . 35 Give a combinatorial argument that the number of partitions of {1, . . , n} such that no two consecutive numbers appear in the same block is precisely the Bell number Bell(n − 1).

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A Course in Enumeration by Martin Aigner

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