Math 5705: Enumerative Combinatorics, Fall 2018: Homework 3

Transcription

1 Uiversity of Miesota, School of Mathematics Math 5705: Eumerative Combiatorics, all 2018: Homewor 3 Darij Griberg October 15, 2018 due date: Wedesday, 10 October 2018 at the begiig of class, or before that by or cavas. Please solve at most 4 of the 6 exercises! 1 Exercise Problem Let A ad B be two sets, ad let f : A B be a map. A left iverse of f shall mea a map g : B A such that g f id A. We say that f is left-ivertible if ad oly if a left iverse of f exists. (It is usually ot uique.) Assume that the sets A ad B are fiite. (a) If the set A is oempty, the prove that f is left-ivertible if ad oly if f is ijective. 1 (b) Assume that f is ijective. Prove that the umber of left iverses of f is A B A. 1.2 Solutio 1 This holds eve whe A ad B are ifiite. eel free to prove this if you wish. 1

2 Solutios to homewor set #3 page 2 of 5 2 Exercise Problem Let A ad B be two sets, ad let f : A B be a map. A right iverse of f shall mea a map h : B A such that f h id B. We say that f is right-ivertible if ad oly if a right iverse of f exists. (It is usually ot uique.) Assume that the sets A ad B are fiite. (a) Prove that f is right-ivertible if ad oly if f is surjective. 2 (b) Prove that the umber of right iverses of f is f 1 (b). Here, f 1 (b) deotes the set of all a A satisfyig f (a) b. b B 2.2 Solutio 3 Exercise Problem (a) Prove that ( ) 1/2 ( 1 4 ) ( ) 2 for each N. (b) Prove that 0 ( 2 )( ) 2 ( ) 4 for each N. [Hit: Part (b) is highly difficult to prove combiatorially. Try usig part (a) istead.] 3.2 Solutio 2 This holds eve whe A ad B are ifiite, if you assume the axiom of choice. But this is ot the subject of our class. Darij Griberg, dgriber@um.edu

3 Solutios to homewor set #3 page 3 of 5 4 Exercise Problem Recall oce agai the iboacci sequece (f 0, f 1, f 2,...), which is defied recursively by f 0 0, f 1 1, ad f f 1 + f 2 for all 2. (1) It is easy to see that f 1, f 2, f 3,... are positive itegers (which will allow us to divide by them soo). ( ) or ay N ad Z, defie the ratioal umber (a slight variatio o the correspodig biomial coefficiet) by ( ) f f 1 f +1, if 0; f f 1 f 1 0, otherwise. (a) Let be a positive iteger, ad let N be such that. Prove that ( ) ( ) ( ) 1 1 f +1 + f 1 1 where we set f 1 1. ( ) (b) Prove that N for ay N ad N., 4.2 Solutio 5 Exercise Problem Let j N, r R ad s R. Prove that j ( )( ) j r s ( 1) s j. j 0 [Hit: irst, argue that it suffices to prove this oly for s N ad r Z satisfyig r sj. Next, cosider r distict stoes, sj of which are arraged i j piles cotaiig s stoes each, while the remaiig r sj stoes are formig a separate heap. How may ways are there to pic j of these r stoes such that each of the j piles loses at least oe stoe?] 5.2 Solutio Darij Griberg, dgriber@um.edu

4 Solutios to homewor set #3 page 4 of 5 Let N. The summatio sig 6 Exercise Problem shall always stad for a sum over all subsets I of []. (This sum has 2 addeds.) Let A 1, A 2,..., A be umbers or polyomials or square matrices of the same size. (Allowig matrices meas that A i A j is ot ecessarily equal to A j A i, so beware of usig the biomial formula or similar idetities!) (a) Show that ( 1) I ( i I A i ) m (i 1,i 2,...,i m) [] m ; {i 1,i 2,...,i m}[] A i1 A i2 A im for all m N. (Example: If 2 ad m 3, the this is sayig (A + B) 3 A 3 B AAB + ABA + ABB + BAA + BAB + BBA, where we have reamed A 1 ad A 2 as A ad B.) (b) Show that ( 1) I ( i I A i ) m 0 for all m N satisfyig m <. (Example: If 3 ad m 2, the this is sayig (A + B + C) 2 (A + B) 2 (A + C) 2 (B + C) 2 + A 2 + B 2 + C , where we have reamed A 1, A 2, A 3 as A, B, C.) (c) Show that ( ) ( 1) I A i A σ(1) A σ(2) A σ(), i I σ S where S stads for the set of all (!) permutatios of []. (Example: If 3, the this is sayig (A + B + C) 3 (A + B) 3 (A + C) 3 (B + C) 3 + A 3 + B 3 + C ABC + ACB + BAC + BCA + CAB + CBA, where we have reamed A 1, A 2, A 3 as A, B, C.) [Hit: You ca use the product rule, which says the followig: Darij Griberg, dgriber@um.edu

5 Solutios to homewor set #3 page 5 of 5 Propositio 6.1 (Product rule). Let m ad be two oegative itegers. Let P u,v, for all u [m] ad v [], be umbers or polyomials or square matrices of the same size. The, (P 1,1 + P 1,2 + + P 1, ) (P 2,1 + P 2,2 + + P 2, ) (P m,1 + P m,2 + + P m, ) P 1,i1 P 2,i2 P m,im. (i 1,i 2,...,i m) [] m (This frighteig formula merely says that a product of sums ca be expaded, ad the result will be a sum of products, with each of the latter products beig obtaied by multiplyig together oe added from each sum. You have probably used this sometime already.) ] 6.2 Solutio Darij Griberg, dgriber@um.edu