Sets with a Negative Number of Elements

Transcription

1 Sets with a Negative Number of Elemets D. Loeb November 16, Itroductio Give a uiverse of discourse U a multiset ca be thought of as a fuctio M from U to the atural umbers N. I this paper, we defie a hybrid set to be ay fuctio from the uiverse U to the itegers Z. These sets are called hybrid sice they cotai elemets with either a positive or egative multiplicity. Our goal is to use these hybrid sets as if they were multisets i order to adequately geeralize combiatorial facts which are true classically for oly oegative itegers. However, the defiitio above does ot tell us much about these hybrid sets; we must defie operatios o them which provide us with the eeded combiatorial structure. We will defie a aalog of a set which ca cotai either a positive or egative umber of elemets. We will allow sums to be calculated over a arbitrary hybrid set. This will lead us to a geeralizatio of symmetric fuctios to a egative umber of variable which agrees with all previous ow geeralizatios of symmetric fuctios i this directio. From these ew symmetric fuctios, we calculate geeralized biomial coefficiets. The these coefficiets will be give a combiatorial iterpretatio i terms of a ew partial order o the hybrid sets. Next, we will geeralize liear partitios usig this partial order ad eumerate them. Fially, we defie sum from oe to a arbitrary iteger ad derive formulas ivolvig them icludig a geeralizatio of the Stirlig umbers. 2 Hybrid Sets Oe usually this of a multiset M as a fuctio from some uiverse U to the atural umbers N. M(x) is called the multiplicity of x i the multiset M. Sets the are a special case of multisets i which all x have multiplicity either 1 or 0. I this paper, we geeralize these cocepts to ew sets ad ew multisets which may have egative itegers as multiplicities as well as oegative itegers. 1

2 2 3 SUMS AND PRODUCTS OVER A HYBRID SET Defiitio 2.1 (Hybrid Sets) Give a uiverse U, ay fuctio f : U Z is called a hybrid set. The value of f(x) is said to be the multiplicity of the elemet x. If f(x) 0 we say x is a member of f ad write x f; otherwise, we write x f. Defie the umber of elemets #f to be the sum x U f(x). f is said to be a #f (elemet) hybrid set. We will deote hybrid sets by usig the usual set braces {} ad isertig a bar i the middle { }. Elemets occurig with positive multiplicity are writte o the left of the bar, ad elemets occurig with egative multiplicity are writte o the right. Order is completely irrelevat. We will ever have occasio to write the same elemet o both sides of the bar. For example, if f = {a, b, c, b d, e, e} the f(a) = 1, f(b) = 2, f(c) = 1, f(d) = 1, f(e) = 2, ad f(x) = 0 for x a, b, c, d, e. Defiitio 2.2 (New Sets) A positive or classical set is a hybrid set taig oly values 0 ad 1. A egative set is a hybrid set taig oly values 0 ad 1. A ew set is either a positive or a egative set. For example, the set S = {a, b, c} ca be writte as the classical or positive set S = {b, a, c } while S = { a, b, c} is a egative set. Both S ad S are ew sets. However, {a b} is ot a ew set. The empty set = { } is the uique hybrid set for which all elemets have multiplicity zero. It is thus simultaeously a positive set ad a egative set. 3 Sums ad Products over a Hybrid Set Oe ofte wishes to sum a expressio over all the elemets of a set. For example, we usually write φ() = i i F() for the Euler phi fuctio where F() is the set of positive factors of a iteger. Whe dealig with multisets, it is ofte coveiet to iclude multiplicities i the summatio. Oe defies x M F(x) to be x U M(x)F(x). Similarly, we ca expad the characteristic polyomial of a matrix as (x λ) where E is its multiset of eigevalues. λ E I geeral, for ay fuctio G ad hybrid set f we write x f G(x) for x U f(x)g(x), ad we write x f G(x) for x U G(x)f(x). As a applicatio, we demostrate that the elemetary symmetric fuctio i a egative umber of variables is i fact the complete symmetric fuctio.

3 3 Defiitio 3.1 (The Complete ad Elemetary Symmetric Fuctios) 1. The complete symmetric fuctio of degree over the set of variable V h (V ) is defied explicitly by the sum h (V ) = x M x M where the sum is over all elemet multisets based o the set of variables V, ad implicitly by the geeratig fuctio xy) x V(1 1 = h (V )y The elemetary symmetric fuctio of degree over the set V of variables e (V ) is defied explicitly by the sum e (V ) = x, S V x S #S= ad implicitly by the geeratig fuctio x V(1 + xy) = 0 e (V )y. Now, if V is the egative set of variables { y 1, y 2,...}, the (1 xt) 1 h (V )t = 0 x V = ( (1 y t) 1) 1 1 = 0 e ( y 1, y 2,...)t, ad e (V )t = (1 + xt) 0 x V = (1 + y t) 1 1 = 0 h ( y 1, y 2,...)t. Thus, e (a, b, c,... x, y, z,...) = h ( x, y, z,... a, b, c,...), so it is immediate that the trasformatio : e h used i the theory of symmetric fuctios is a ivolutio.

4 4 4 ROMAN COEFFICIENTS 4 Roma Coefficiets Whe V is a classical set of variables, e (V ) cotais ( terms, so if we set all the variables to oe, the we have the biomial coefficiet: ( ) e (V ) x=1 =. x V However, the left had side of this equatio is well defied eve whe is a egative iteger ad V is a egative set. I this case, it ca be used as a geeralizatio of the biomial coefficiets. This geeralizatio is already well ow, ad it is usually ( ) = ( 1)( 2) ( + 1).! However, this is oly well defied for oegative; usig the followig result we will permit both ad to be arbitrary itegers. Theorem 4.1 For all itegers ad all oegative itegers, ( ) Γ( ǫ) = lim ǫ 0 Γ( ǫ)γ( ǫ). (1) We will adopt Equatio 1 as the defiitio of the biomial coefficiets eve whe is egative. Table 4.1 (Biomial Coefficiets, ( ) \ Propositio 4.1 (The Six Regios) Let ad be itegers. Depedig o what regio of the Cartesia plae (, is i, the followig formulas apply:

5 5 1. If 0, the ( ( =. 2. If 0 >, the ( ) ( ) = ( 1) If 0 >, the ( ) ( ) = ( 1) If > 0, ( = If 0 > >, ( = If 0 >, ( = 0. Note that i regios 4 6 there is a extra factor of ǫ i the umerator of the limit, so we are left with zero. Next, regio 1 is the classical case, so we have the usual biomial coefficiets. Most of the usual properties of biomial coefficiets hold true i all six regios. Propositio 4.2 (Complemetatio) For all itegers ad m, ( ) ( m = m). Propositio 4.3 (Iteratio) For all itegers i, j, ad, ( )( ) ( )( ) i j i i j =. j j Propositio 4.4 (Pascal) Let ad be itegers ot both zero, the ( ) ( ) ( ) 1 1 = +. 1 Nevertheless, 0 0 = 1 while = = 2. Moreover, it is possible to prove a geeralizatio of the biomial theorem via Gamma-Coefficiets: Propositio 4.5 (Gamma Theorem) For all itegers, 1. The coefficiets of the formal power series (x + 1) are give by [x ](x + 1) = ( where is a oegative iteger. 2. The coefficiets of the iverse power series (x + 1) are give by ( ) ( ) [x ](x + 1) = = where is a egative iteger. Proof: Propositio 4.4 ad iductio.

6 6 5 INCLUSION OF HYBRID SETS 5 Iclusio of Hybrid Sets These ew biomial coefficiet ( are always itegers, but what do they cout. I regio 1, they cout the umber of elemet subsets of a give elemet set. We claim that give a suitable geeralizatio of the otio of a subset, this is true i geeral (up to sig 1 ). We defie a partial order o these sets which is a geeralizatio of the usual orderig of classical sets ad multisets by iclusio; we will give the defiitio twice: First, a iformal motivatio ad the a formal defiitio. Iformally, we say f is a subset of g (ad write f g) if oe ca remove elemets oe at a time from g (ever removig a elemet that is ot a member of g) ad thus either acheive f or have removed f. For example, we might start with the hybrid set f = {a, b, c, c d, e}. We will remove a few of its elemets oe at a time. Suppose we remove b, this leaves {a, c, c d, e}. Now, b is o loger a elemet so we ca ot remove it agai. Istead, we might remove d leavig {a, c, c d, d, e}. Obviously, we ca remove d as may times as we chose. Fially, suppose we remove c leavig {a, c d, d, e}. Hece, we have prove two thigs. Sice we were able to remove {b, d, c }, we ow that {b, d, c } f. Also, sice we were left with {a, c d, d, e}, we ow that {a, c d, d, e} f. Defiitio 5.1 (Subsets) Let f ad g be hybrid sets. We say that f is a subset of g ad that g cotais f ad we write f g if either f(x) g(x) for all x U, or g(x) f(x) g(x) for all x U where is a partial orderig of the itegers defied as follows: i j if ad oly if i j ad either i < 0 or j 0. Note that the Hasse diagram of the relatio is obtaied from the usual Hasse diagram of the itegers by discoectig 0 from 1; it cosists of the disjoit uio of a ascedig chai (the oegative itegers), ad a descedig chai (the egative itegers). Propositio 5.1 The orderig defied above is a well defied partial order. Proof: Trasitivity is the oly property worth checig. Suppose f g ad g h. We must prove that f h i each of the followig three cases. Note first that by the above remars is a partial order. 1. Suppose f g ad g h. Sice is a partial order, f h. 2. Suppose f g ad h g h. The g is a legal set of removals from h. g must be a ordiary multiset, so by Theorem 5.1, f is a smaller multiset, ad must thereof also be a legal set of removals. Hece f h.. 1 The sig is give by e (V ) x=1 x V ( ) =.

7 7 3. Suppose g f g ad g h. Now, f is a legal set of removals from g. However, g is the remaider of h after aother series of removals, so f is a legal set of removals from h. Note however that this orderig does ot form a lattice. For example, hybrid sets f = { a, b} ad g = {a b} have lower bouds {a }, {a, b }, ad {a, b, b }, but o greatest lower boud. Coversely, f ad g possess o upper bouds. Theorem 5.1 The subsets of a classical set f correspod to the classical subsets of a classical set. The subsets of a multiset f correspod to the classical submultisets of a multiset. Proof: To costruct a covetioal subset of a set S, we merely remove some elemets subject to the coditios that we oly remove elemets of S ad we do t remove a elemet twice. The order of removal is ot relevat. To costruct a covetioal submultiset of a multiset M, we merely remove some elemets subject to the coditios that we do ot remove ay elemet more times tha its multiplicity i M. Now, we have the desired iterpretatio of the biomial coefficiets. Theorem 5.2 Let ad be arbitrary itegers. Let f be a -elemet ew set. The ( couts the umber of -elemet hybrid sets which are subsets of f Let us cosider this result i each of the six regios of Propositio This is the oly classical case. I this regio, oe might cout the umber of 2 elemets subsets of the set {a, b, c, d }. By Theorem 5.1, we eumerate the usual subsets: {a, b }, {a, c }, {a, d }, {b, c }, {b, d }, ad {c, d } but o others. 2. I this regio, oe might cout the umber of 2 elemet subsets of f = { a, b, c}. These subsets correspod to what we ca remove from f, sice what we would have left over after a removal would ecessarily cotai a egative umber of elemets. We ca remove ay of the three elemets ay umber of times, so we have: {a, a }, {b, b }, {c, c }, {a, b }, {b, c }, ad {a, c }. 3. Here we are iterested i -5 elemet subsets of f. Sice f cotais -3 elemets, we must start with f ad remove 2 elemets. Thus, there is o subset here for each subset i the correspodig positio i regio 2. I this case they are: { a, a, a, b, c}, { a, b, b, b, c}, { a, b, c, c,c}, { a, a, b,b, c}, { a, b, b, c, c}, ad { a, a, b, c, c}. 4. By Theorem 5.1, there are o 6 elemets subsets of the set {a, b, c, d }. Oce you remove 4 elemets, you ca ot remove aymore. 5. There are o -2 elemet subset of a -3 elemet set f. If we remove elemets from f, we are left with less tha -3 elemets, ad have removed a positive umber of elemets. I either case have we qualified a -2 elemet hybrid set to be a subset of f. 6. Agai by Theorem 5.1, there are o -2 elemet submits of the set {a, b, c, d }, sice you are ot allowed to itroduce elemets with a egative multiplicity.

8 8 6 SUMS AND PRODUCTS WITH LIMITS 6 Sums ad Products with Limits Oe ofte uses the otatio i =1 A() where is a oegative iteger to deote the sum of A() over the set {1, 2,..., }. I aalogy to the situatio with itegrals, we will defie the sum or product of a quatity from a arbitrary iteger to aother. Defiitio 6.1 (Sums ad Products with Limits) We recursively defie the sum or product of A() from equals i to j j j A() or A() by two coditios. First, ad secod, i A() = 0 j+1 A() = A(j + 1) + j A() ad ad i A() = 1, j+1 A() = A(j + 1) j A(). Just as with itegrals, the sum is shift ivariat, the sum over a empty iterval is zero, ad the sum over a positive expressio over a bacwards iterval j < i is egative. The sum or product from i to j ca be thought of as the sum or product over the set {i..j} where {i, i + 1,..., j 1, j } if i < j, {i..j} = if i = j, ad { j + 1, j + 2,...,i 1} if i > j. For example, 4 i=1 i = = 10, ad 4 i=1 = ( 3) ( 2) ( 1) 0 = 6. I geeral, i=1 i = (+1) 2 for regardless of sig. I fact, Propositio 6.1 Let p(x) be ay polyomial, the there is a polyomial q(x) such that q() = i=1 p(i) for all regardless of sig. Proof: Classical proof by iductio ca be applied here mutatis mutadus. As a applicatio of this otio of a iterval, ad the duality betwee the elemetary ad complete symmetric fuctios metioed earlier, cosider the Stirlig umbers of the first id. Defiitio 6.2 (Stirlig Numbers of the First Kid) Let be a iteger, ad let be a oegative iteger. The the Stirlig umber of the first id of degreee ad order s(, is defied to be the coefficiet of x i the Taylor series expasio of (x) = 1 i=0 (x i). The we have the followig beautiful sythesis Theorem 6.1 For all itegers (positive or egative) ad oegative itegers, ( ) 1 s(, = lim Γ(1 + ǫ) 1 e ǫ 0 ǫ, ǫ, ǫ,..., 1. + ǫ

9 9 7 Liear Partitios Just as oe ca more carefully study the combiatorial properties of the biomial coefficiets by itroducig the Gaussia coefficiet, here we study the geeralized Gaussia coefficiet. Classically, oe defies the Gaussia coefficiet ( ) q We geeralize the Gaussia coefficiet as follows = (q 1)(q 1 1) (q +1 1) (q 1)(q 1. 1) (q 1) Defiitio 7.1 (Gausssia Coefficiet) Let be a iteger ad be a oegative iteger. The defie the Gaussia coefficiet ( ) q +1 i 1 i=1 = q (q 1)(q 1 1) (q 1). We will iterpret these Gaussia coefficiets i terms of liear partitios. Oe usually defies a (liear) partitio to be a fiite oicreasig sequece of oegative itegers. Sice we are ow givig egative itegers a status equal to that of oegative oes, this defiitio is iappropriate. Defiitio 7.2 (New Partitio) If λ is a fiite sequece of itegers of legth such that λ i λ i+1 for all 1 i <, the we say that λ is a ew partitio. The λ i are called the parts of λ, ad we defie λ to be the sum of the parts of λ. We say that λ is a partitio of λ For example, (5, 3, 2, 2, 0) ad ( 3, 7, 7, 32) are ew partitios, but (3, 1, 0, 1) is ot. It is well ow (see for example Kuth) that for ad oegative, ( ) is a moic polyomial q of degree. Its coefficiet of q t is the umber of partitios λ of t of legth with all parts less tha or equal to. I geeral, Theorem 7.1 Let be a oegative iteger, ad ad t be arbitrary itegers. The ( ) = q t c tq t where c t is the umber of partitios λ of t of legth with all parts. 8 Fuctios We ow defie the hybrid sets of fuctios ad ijectios betwee a classical set S ad a arbitrary hybrid set A. Defiitio 8.1 (Fuctios) Give a classical set S, ad a hybrid set A. Defie Fu(S, A) ad Moo(S, A) to be hybrid sets of fuctios from S to the uiverse U. The multiplicity of f : S U i Fu(S, A) is x S A(f(x)). Whereas, the multiplicity of f i Moo(S, A) is ( A(x) x U #f (x)) where #f 1 (x) is the size 1 of the iverse image of x uder f f 1 (x). For example, if A is a classical set the Fu(S, A) is the set of fuctios from S to A, ad Moo(S, A) is the set of such ijectios. Here we see that #Fu(S, A) = #A (#S), ad cofirm Staley s formula that Ω(P, ) couts poset homomorphism oto chais, if Ω(P, ) couts strict poset homomorphisms oto chais.

10 10 REFERENCES 9 Other Applicatios Sice submittig this article, the author has read about surreal umbers [1] which ca be iterpretted as a example of this theory. Aother applicatio (cocerig the coectio costats betwee polyomial sequeces ad/or iverse formal power series sequeces) will be appear i a separate article. [2] Refereces [1] J. W. Coway, O Numbers ad Games, Academic Press, Lodo, [2] E. Damiai, O. D Atoa, ad D. Loeb, The Complimetary Symmetric Fuctio: Coectio Costats Usig Negative Sets, To appear. [3] D. Kuth, Subspaces, Subsets, ad Partitios, Joural of Combiatorial Theory 10 (1971) [4] D. Loeb, A Geeralizatio of the Biomial Coefficiets, SIAM Joural of Discrete Mathematics, To appear. [5] D. Loeb, A Geeralizatio of the Stirlig Numbers, SIAM Joural of Discrete Mathematics, To appear. [6] D. Loeb ad G.-C. Rota, Formal Power Series of Logarithmic Type, Advaces i Mathematics, 75 (1989), [7] I. G. Macdoald, Symmetric Fuctios ad Hall Polyomials, Oxford Mathematical Moographs, Claredo Press, Oxford, [8] S. Roma, A Geeralizatio of the Biomial Coefficiets, To Appear.