New PDF release: A Course in Enumeration

By Martin Aigner

ISBN-10: 3540390324

ISBN-13: 9783540390329

Combinatorial enumeration is a simply available topic jam-packed with simply acknowledged, yet occasionally tantalizingly tough difficulties. This booklet leads the reader in a leisurely means from the elemental notions to numerous themes, starting from algebra to statistical physics. Its objective is to introduce the scholar to a fascinating box, and to be a resource of knowledge for the pro mathematician who desires to examine extra in regards to the topic. The ebook is geared up in 3 components: fundamentals, tools, and subject matters. There are 666 workouts, and as a distinct function each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.

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Example text

If we fix the domain {1, 2, . . , 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.

47 Prove the following identities for the Stirling numbers sn,k : a. n n−i i=k si,k n = n! k n si,k i! i=k k−m c. k sn+1,k+1 m (−1) ( , m, n ∈ N0 ). n i=0 (m = sn+1,k+1 , b. = sn,m , d. 48 Let σ = a1 a2 . . an ∈ S(n) be given in word form. A run in σ is a largest increasing subsequence of consecutive entries. The Eulerian number An,k is the number of σ ∈ S(n) with precisely k runs or equivalently with k − 1 descents ai > ai+1 . , An,1 = An,n = 1 with 12 . . n respectively n n − 1 . . 1 as the only permutations.

40 Let in be the number of permutations of {1, . . , n} with no cycles (r ) (r ) n of length greater than r . Prove the recurrence in+1 = k=n−r +1 nn−k ik , generalizing the previous exercise. 41 Let > 2 . Show that the number of permutations σ ∈ S(n) that n! have a cycle of length equals . What is the proportion t(n) of σ ∈ n S(n) that contain a cycle of length > 2 when all permutations are equally likely? Compute limn→∞ t(n). 42 Let In,k be the number of permutations in S(n) with exactly k inn 2 .

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A Course in Enumeration by Martin Aigner

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