By Martin Aigner

Combinatorial enumeration is a with no trouble obtainable topic packed with simply said, yet occasionally tantalizingly tricky difficulties. This publication leads the reader in a leisurely method from the fundamental notions to quite a few issues, starting from algebra to statistical physics. Its goal is to introduce the scholar to a fascinating box, and to be a resource of data for the pro mathematician who desires to study extra in regards to the topic. The ebook is prepared in 3 elements: fundamentals, tools, and subject matters. There are 666 routines, and as a different characteristic each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.

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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). 36 Show that (Sn,0 , Sn,1 , . . , Sn,n ) is a unimodal sequence, for every n. More precisely, prove that there is an index M(n) such that 24 1 Fundamental Coeﬃcients Sn,0 < Sn,1 < · · · < Sn,M(n) > Sn,M(n)+1 > · · · > Sn,n or Sn,0 < Sn,1 < · · · < Sn,M(n)−1 = Sn,M(n) > · · · > Sn,n , where M(n) = M(n − 1) or M(n) = M(n − 1) + 1.

We have p(i; ≤ n; ≤ m)qi = Am,n (q) = i≥0 m+n m q . 6 Lattice Paths and Gaussian Coeﬃcients 39 m+n Proof. We show that Am,n (q) and [ m ]q satisfy the same initial conditions and the same recurrence. For m = 0 or n = 0 we have A0,n (q) = Am,0 (q) = 1 since in this case we get only the empty m+0 partition, and also [ 0+n 0 ]q = [ m ]q = 1. Assume m, n ≥ 1; then m+n the recurrence for [ m ]q is by (1), m+n m q m+n−1 m−1 = q m+n−1 m + qm q . (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,k = In,(n)−k , c. In,k = 2 (n2 ) In−1,k + In,k−1 for k < n. Is this also true for k = n? d. k=0 (−1)k In,k = 0 for n ≥ 2. The In,k are called inversion numbers. versions, k = 0, 1, . . 43 Let σ = a1 a2 . . an ∈ S(n) be given in word form, and denote by bj the number of elements to the left of j that are larger than j (thus they form an inversion with j). The sequence b1 b2 . . bn is called the inversion table of σ . Show that 0 ≤ bj ≤ n − j (j = 1, . . , n) and prove, conversely, 30 1 Fundamental Coeﬃcients that every sequence b1 b2 .