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Mathematics 501 Homework (due Nov 1) A. Hulpke 43) Show that there are exactly Cn Young diagrams that “fit in the shape” (n − 1, n − 2, . . . , 1). 44) Using the Hook formula, determine the number of tableaux of the shape 71 51 32 1. 45) a) Show that if a diagram has a hook of length a + b it must have a hook of length a or of length b. (Hint: Consider a zig-zag path between the ends of the hook that goes around the bottom edge of the diagram) b) Show that if a diagram has a hook of length ab it must have hooks of length a and length b. 46) Using the RSK correspondence, determine the tableaux corresponding to the permutation (1, 7)(3, 8, 6, 4, 5). Determine a permutation corresponding to the following pair 1 4 8 1 3 8 2 6 2 4 3 7 5 7 5 and 6 . of tableaux: 47) Let s(n) be the number of permutations on n points for which π 2 = 1. a) Show that s(n) = s(n − 1) + (n − 1)s(n − 2) b) Show that the exponential generating function for s(n) is tn 2 ∑ s(n) n! = et+t /2 n (Hint: Consider the differential equation f 0 (t) = (1 + t) f (t).) 48) Let f (x) = Q[x] be a monic (i.e the coefficient of the largest power of x occurring is 1) polynomial. We define the discriminant of f as D( f ) = ∆2 ( f ) = ∏(αi − α j ) i6= j where the {αi } are the roots of f , including multiplicities. a) Show that D( f ) = 0 if and only f has multiple roots. b) Show that if f (x) ∈ Z[x] then D( f ) ∈ Z. c) Show that if f (x) ∈ Z[x] has no multiple roots that there are only finitely many primes p such that f has multiple roots modulo p.