Introduction To Discrete Mathematics

Transcription

1 Itroductio To Discrete Mathematics Review If you put + pigeos i pigeoholes the at least oe hole would have more tha oe pigeo. If (r + objects are put ito boxes, the at least oe of the boxes cotais r or more of the objects. If the average of oegative itegers a,a 2, a is greater tha r, i.e., a + a a > r, the at least oe of the itegers is greater tha or equal to r. The umber of r-permutatios of a -set equals P(,r ( ( r + The umber of permutatios of a -set is P(,!. The umber of circular r-permutatios of a -set equals! ( r!. P(,r r! ( r!r. The umber of circular permutatios of a -set is equal to (! The umber of r-combiatios of a -set equals ( P(,r! r r! ( r!r!. The umber of r-permutatios of the multiset { x, x 2,, x } equals r. The umber of permutatios of the multiset { x, 2 x 2,, x } equals!! 2!!, where The the umber of r-combiatios of the multiset { a, a 2,, a } (the umber of r-combiatios with repetitio allowed equals ( +r ( r +r. The ( umber of oegative iteger solutios for the equatio x + x x r equals +r ( r +r. The umber of positive iteger solutios for the equatio x + x x r equals ( r. The umber of ways to place r idetical balls ito distict boxes equals ( ( +r r +r. The umber of ways to place r idetical balls ito distict boxes such that o box remais empty equals ( r.

2 Algorithm for geeratig the permutatios of {, 2,,, }: Begi with 2. While there exists a mobile iteger, do ( Fid the largest mobile iteger m (2 Switch m ad the adjacet iteger its arrow poits to. (30 Switch thew directio of all the arrows above itegers p with p > m. Algorithm for costructio of a permutatio from its iversio sequece (a,a 2,,a : ( Write dow. (- Isert to the right of the a th existig umber Algorithm 2 for costructio of a permutatio from its iversio sequece (a,a 2,,a : (0 Mar dow empty spaces. For till Put ito the a + st empty space from the left. Algorithm for geeratig combiatios of {x,x 2,,x,x 0 } : Begi with a a 2 a a While a a 2 a a 0, do ( Fid the smallest iteger j such that a j 0. (2 Replace a j by ad each of a j,,a,a 0 by 0. The algorithm stops whe a a 2 a a 0. Algorithm for geeratig reflected Gray codes of order : Begi with a a 2 a a While a a 2 a a , do ( If a + a a + a 0 eve, the chage a 0 (from 0 to or to 0. (2 If a + a a + a 0 odd, fid the smallest j such that a j ad chage a j+ (from 0 to or to 0. Algorithm for geeratig r-combiatios of S {, 2,,, }: Begi with 2 r. While a a 2 a r ( r + (, do ( Fid the largest iteger such that a < ad a + is ot i the a a 2 a r. (2 Replace a a 2 a r with a a 2 a (a + (a + 2 (a + r +. Algorithm for a liear extesio of a -poset: Step. Choose a miimal elemet x from X (with respect to the orderig. Step 2. Delete x from X; choose a miimal elemet x 2 from X. Step 3. Delete x 2 from X ad choose a miimal elemet x 3 from X. Step. Delete x from X ad choose the oly elemet x i X.

3 For a real α ad a iteger, ( α α(α (α +! if if 0 0 if. ( ( ( + ( ( ( (0 ( ( ( ( ( ( ( ( ( ( ( ( + + ( 0 ( 0 2 ( ( ( ( ( 2 ( ( ( ( ( 2 ( ( ( ( ( 2 ( 2 ( 2 ( 2 ( 2 ( ( ( ( ( ( ( ( ( ( ( ( ( + Biomial expasio. For iteger ad variables x ad y, (x + y ( x y. Newto s biomial expasio. For a real α ad variables x ad y with 0 x y, (x + y α ( α x y α. Multiomial expasio. For iteger ad variables x,x 2,,x, ( (x + x x t x, 2,, x 2 2 xt t. t t;, 2,, t 0

4 Sperer s theorem. Ay clutter of a -set S cotais at most ( 2 subsets of S. The power set P(S ca be partitioed ito m disjoit chais C,C 2,,C ( 2. Costructio of a symmetric chai partitio for the case give a symmetric chai partitio for the case : for each chai A A 2 A for the case : if 2, do A A 2 A A {} ad A {} A 2 {} A {}; if, do A A 2 A A {}. Dilworth s theorem. mi{ : A A is a atichai partitio } max{ C : C is a chai }. mi{ : C C is a chai partitio } max{ A : A is a atichai }. Let P,P 2,,P be properties of the objects of a fiite set S. Let A i be the set of all elemets of S that have the property P i. The umber of objects of S that have oe of the properties P,P 2,,P is give by Ā Ā2 A S A i + A i A j A i A j A + +( A A 2 A. i i<j i<j< The umber of objects of S that have at least oe of the properties P,P 2,,P is give by A A 2 A A i A i A j + A i A j A +( + A A 2 A. i i<j i<j< A permutatio i i 2 i of {,2,,} is called a deragemet if i for ay (o umber remais i its positio.the umber D of deragemets of {,2,,} is give by D!(! + 2! 3! + + (!. The deragemet sequece D satisfies the followig recurrece relatios D ( (D + D 2, D 0,D 2, ad D D + (, D 0. A permutatio of {,2,,} is called ocosecutive if oe of 2,23,,( occurs. The umber Q of ocosecutive permutatios of {,2,,} is give by ( Q ( (! For 2, Q D + D. A circular permutatio of {,2,,} is called ocosecutive if oe of 2,23,, occurs. The umber C of ocosecutive circular permutatios is give by ( C ( (! + (. Let X m ad let Y. The umber of all fuctios from X to Y equals m. The umber of ijective fuctios from X to Y equals ( m m! P(,m. The umber S(m, of surjective fuctios from X to Y is give by ( S(m, ( ( m.

5 Theorem. Let q 0. The geometric sequece h q is a solutio of the recurrece relatio h a h + a 2 h a h ; a 0, ( if ad oly if the umber q is a root of the characteristic equatio x a x a 2 x 2 a x a 0 0. (2 Theorem 2. If the characteristic equatio (2 has distict roots q,q 2,,q, the the geeral solutio of ( is h c q + c 2 q c q. Theorem 3. Let q,q 2,,q s be distict roots with the multiplicities m,m 2,,m s respectively for the characteristic equatio (2. The the sequeces q,q, 2 q,, m q ; q 2,q 2, 2 q 2,, m 2 q 2; q s,q s,2 q s,,ms q s ; are liearly idepedet solutios of the recurrece relatio (. Their liear combiatios form the geeral solutio of the recurrece relatio (. Theorem 4. Let h be ay particular solutio of the recurrece relatio h a h + a 2 h a h + b ; a 0,, (3 ad let h be the geeral solutio of its correspodig the homogeeous recurrece relatio. The h h + h is the geeral solutio of the recurrece relatio (3. Cosider a first-order liear ohomogeeous recurrece relatio h ah + b ; (4 Theorem 5. Let b cq. The (4 has a particular solutio of the followig form: If q a, the h Aq. If q a, the h Aq. Theorem 6. Let b i0 c i i. If a, the (4 has a particular solutio of the form h A 0+A +A A. If a, the the solutio of (4 is h h 0 + i0 b i, Theorem 7. Give a ohomogeeous liear recurrece relatio of the secod order h a h + a 2 h 2 + cq ; 2 (5 Let q ad q 2 be solutios of its characteristic equatio x 2 a x a 2 0. The (6 has a particular solutio of the followig forms: If q q,q q 2, the h Aq. If q q,q q 2, the h Aq.

6 If q q q 2, the h A 2 q. Theorem 8. Give a ohomogeeous liear recurrece relatio of the secod order where b is a polyomial fuctio of with degree. h a h + a 2 h 2 + b ; 2 (6 If a + a 2, the (6 has a particular solutio of the form: h A 0 + A + A A, where the coefficiets A 0,A,,A are to be determied. If 2, the a particular solutio has the form h A 0 + A + A 2 2. If a + a 2, the (6 ca be reduced to a first order recurrece relatio g (a g + b, where g h h for. For the sequece a 0,a,a 2,,a,, its ordiary ad expoetial geeratig fuctios are give by A(x a 0 + a x + a 2 x a x + A (e x (x a 0 + b! + a x 2 2 2! + + a x! + A(xB(x A (e (xb (e (x ( a i b i x i0 ( ( a i b i x i! i0 Some ordiary geeratig fuctios: a i c i i i 2 ( i A(x x cx x ( x 2 ( +i i i ;a 0 0 x(+x ( x 3 ( + x ( x l x Some expoetial geeratig fuctios: a i c i i i 2 i! ( i (i A (e (x e x e cx xe x x(x + e x ( x ( + x ( x

7 Give a colorig c C, the stabilazor of c is the set G(c {f G f c c}. Give a permutatio f G, the ivariat set of f is the set C(f {c C f c c}. Give a colorig c C, the orbit of c is the set c {f(c f G}. Let C be the set of all colorigs of X ito colors. The C(f #(f, where #(f is the umber of cycles i the disjoit cycle factorizatio of f. Burside s Lemma Suppose a group G of permutatios of X acts o a set C of colorigs of X. The the umber N(G,C of oequivalet colorigs i C is give by N(G,C C(f. G f G Give a permutatio f G, the type of f is a -tuple type(f (e,e 2,,e, where e i is the umber of i-cycles i a disjoit cycle factorizatio of f. e + e e #(f, e + 2e e. To each permutatio f G with type type(f (e,e 2,,e we associate a moomial The cycle idex of G is mo(f z e ze 2 2 ze P G (z,z 2,,z mo(f z e G G ze 2 2 ze. f G f G Theorem Suppose there are colors. Let C be a set of all colorigs of X. The the umber N(G,C of oequivalet colorigs i C is give by N(G,C P G (,,,. Theorem (Polya Let {u,u 2,,u } be a set of colors. Let C be a set of ay colorigs of X such that the group G of permutatios of X acts o the set C. The the geeratig fuctio for the umber of oequivalet colorigs i C accordig to the umber of colors of each id is give by P G (u + u,u 2 + u 2,,u + u.