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Problem 17E. Let A denote the Latin square of order 10 on th...

Problem 17E. Let A denote the Latin square of order 10 on the left below, and let C be the partial 10 by 10 Latin rectangle on the 11 symbols 0, 1,2, 9, O on the right (not including the last row and column). Use the algorithm of the proof of Theorem 17.5 to complete C to a 10 by 10 Latin rectan...

Problem 14B. Describe (with a figure) a correspondence betwe...

Problem 14B. Describe (with a figure) a correspondence between the graphs of Question 1 and those of Question 2. Question 1. A tree that is drawn in the plane is called a plane tree. How many rooted plane trees are there with n vertices and a root with degree 1 (so-called planted plane trees)? ...

Problem 15E. Show with a Ferrers diagram that the number of ...

Problem 15E. Show with a Ferrers diagram that the number of partitions of n + k into k: parts equals the number of partitions of n into at most k: parts. (This is (15.2).) Many theorems about partitions can be proved easily by repre- senting each partition by a diagram of dots, known as a Ferrer...

Problem 15F. Prove that the number of self-conjugate partiti...

Problem 15F. Prove that the number of self-conjugate partitions of n equals the number of partitions of n into unequal odd parts. The partition we get by reading the Ferrers diagram by columns instead of rows is called the conjugate of the original partition. So

Let p(n) denote the number of partitions of n. so that p(6) ...

Let p(n) denote the number of partitions of n. so that p(6) = 11. Note that p(0) = 1 (either by definition or by interpretation). Let f(x) be the generating function of the sequence p(0),p(1),p(2), p(n). In other words, Show that 1 1 1 f(x) = (1+I+I+ (1+r²+r4+ )(1+2³+r6+...) = Le...

1. The commutator of A and B is [A,B] = AB - BA. Thus A and...

1. The commutator of A and B is [A,B] = AB - BA. Thus A and B commute if and only if [A,B] = 0. (Assume that A and B obey the other usual rules of arithmetic: associativity of multiplication, distributivity, etc.) (a) Show that [A, B] = - [B,A]. (b) Prove the Jacobi identity, [A, [B, C1] + [C,...

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