By Martin Aigner
Combinatorial enumeration is a without difficulty obtainable topic filled with simply acknowledged, yet occasionally tantalizingly tricky difficulties. This booklet leads the reader in a leisurely means from the elemental 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 knowledge for the pro mathematician who desires to study extra concerning the topic. The ebook is equipped in 3 components: fundamentals, tools, and themes. There are 666 workouts, and as a different function each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.
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Extra resources for A Course in Enumeration
What is |λ|=n f1 (λ)? Hint: Show that the sums satisfy the same recurrence. 63 . Let bm be the number of pairs λ = λ1 λ2 . . λk , μ = μ1 μ2 . . μk for some k such that λ1 > λ2 > · · · > λk ≥ 1, μ1 > μ2 > · · · > μk ≥ 0 and λj + μj = m. Prove that bm = p(m). Hint: Use a clever decomposition of the Ferrers diagram. 64 A partition of n is called perfect if it contains precisely one partition for every m < n. Thus, if x1 + 2x2 + · · · + nxn = n (xi ≥ 0), then there is a unique solution of y1 + 2y2 + · · · + nyn = m with 0 ≤ yi ≤ xi for all i and m < n.
Sn,r rn n! , λ∗ = λ, equals the number of partitions of n with all summands odd and distinct. 54 Show that p(n; n − t) = p(t) if and only if n ≥ 2t. 55 Prove pd (n; k) = p(n − k 2 ; k). 56 Verify the recurrence pd (n; k) = pd (n − k; k) + pd (n − k; k − 1), 2 ≤ k ≤ n 2 ,n ≥ 5, with starting values pd (n; 1) = 1, pd (n; k) = 0 for n < k+1 2 , pd k+1 2 ;k = 1. 57 Express the following quantities in terms of Fibonacci numbers: a. The number of ordered partitions of n into parts greater than 1; b. the number of ordered partitions of n into parts equal to 1 or 2; c.
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. Prove the recurrence An,k = (n − k + 1)An−1,k−1 + kAn−1,k for n, k ≥ 1 with A0,0 = 1, A0,k = 0 (k > 0).