A Mean Paradox. John Konvalina, Jack Heidel, and Jim Rogers. Department of Mathematics University of Nebraska at Omaha Omaha, NE USA

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1 A Mea Paradox by Joh Kovalia, Jac Heidel, ad Jim Rogers Departmet of Mathematics Uiversity of Nebrasa at Omaha Omaha, NE USA Phoe: (42) Fax: (42)

2 A Mea Paradox I our departmet we have a mathematical biology group that icludes studet researchers studyig mathematical models ( ow as Boolea etwors) of complex biological systems [4]. Some of these models ca be represeted as permutatios o a fiite set whose size grows expoetially. Oe objective is to compute cycle structure statistics of the permutatios, sice these ofte have some biological relevace. The importat cycle statistics iclude the average or mea umber of cycles i a set of permutatio ad the mea cycle legth. Sice the space of permutatios is so large, oe must ofte resort to simulatios ad samplig, icludig radomly geeratig permutatios ad studyig their cycles. It is well ow from group theory that every permutatio o a fiite set ca be writte as a cycle or a product of disjoit cycles [2]. A classical result from combiatorics ad discrete probability is that the mea cycle legth of permutatios o symbols is approximately / log. However, if we radomly geerate a permutatio of degree, perform the decompositio ad average the cycle legths, ad repeat this process may times, we fid that the overall average or mea cycle legth differs otrivially from the classical result. Why should the mea cycle legth arisig from radom permutatio geeratio of a applied modelig problem differ from the classical mea cycle legth? I this ote we will show there is a surprisigly otrivial resolutio of this mea paradox. First, we prove the classical result that the mea cycle legth is approximately / log. The origi of this proof ca be traced bac at least 5 years to a paper that appeared i this Mothly by R.E. Greewood [3]. Let c deote the mea cycle legth i permutatios of size. The Stirlig umbers of the first id, deoted by coefficiets of the followig polyomial :, arise as F ( x) x( x + )( x + 2) ( x + ) x () The Stirlig umber has a combiatorial iterpretatio as the umber of permutatios of the umbers to with exactly cycles i the cycle decompositio [5]. Thus, summig over yields the total umber of cycles. But this is just the derivative of () evaluated at x. That is, F ( ). 2

3 If we divide F () by F ( )!, the total umber of permutatios, we obtai the mea umber of cycles i a permutatio : F () F() H where H is the th Harmoic umber. It is well ow that umber of cycles over all permutatios is H ~ log. Thus, the total! H. The total of all cycle legths is! sice the sum of the legths of the cycles i a give permutatio is. So the mea cycle legth is c! ~.! H H log Observe that the mea cycle legth i the classical case was actually obtaied by averagig over all the cycles. But, i our simulatios we radomly geerate permutatios ad the averaged the resultig cycle legths. What is the mea cycle legth whe averagig over all permutatios? Let p deote the mea cycle legth over all permutatios. There are permutatios with exactly cycles i the cycle decompositio. The average cycle legth for these permutatios is / sice the total umber of elemets is ad there are cycles. Thus, summig over ad dividig by the total umber of permutatios, we have p! Usig () we obtai a itegral represetatio for the mea cycle legth: p ( )! F( x) x ( )! Or, usig the gamma fuctio Γ (x) : p ( x + )( x + 2) ( x + ) ( )! Γ( x + ) Γ( x + ) Γ( ) 3

4 Next, we will derive a recurrece relatio for p. Observe that p + ( x + )( x + 2) ( x + )( x + )! ( x + )( x + 2) ( x + ) + ( )! x ( x + )( x + 2) ( x + )! x ( x + )( x + 2) ( x + ) p +.! To evaluate the itegral we use a classical result from the Calculus of Fiite Differeces ([6] pp.3, 82): B ( ) x ( x + )( x + 2) ( x + ) ( ) ( ) B ( ) ( where B ) is the th Beroulli umber of order. The expoetial geeratig fuctio for these umbers is ([6] p.35): ( ) ( ) t t B. (2)! ( t)log( t) So the recurrece relatio for p is ( ) B ( ) p+ p + (3)! 4

5 () with p, B. Usig the geeratig fuctio (2) together with (3) we obtai the geeratig fuctio G (t) for p :. 2 t G ( t) pt (4) 2 ( t) log( t) A geeralizatio of the geeratig fuctio (4) was studied by Flajolet ad Odlyzo []. Usig Cauchy s itegral formula ad Hael type cotours, they derived asymptotic expasios for the coefficiets. Applyig their Theorem 3A we obtai the followig asymptotic expasio for the mea cycle legths over the permutatios: p + ~ log a + log where d a. Γ( x) x 2 So the resolutio of the mea paradox has led us to a otrivial asymptotic expasio of the mea cycle legth p from a closed-form expressio of the associated geeratig fuctio. Acowledgemet This research has bee partially supported by NIH grat #R6M672- Refereces. Flajolet, P. ad Odlyzo, A., Sigularity aalysis of geeratig fuctios. SIAM J. Discrete Math. 3 (99) Gallia, J., Cotemporary Abstract Algebra (5 th ed.), Houghto-Miffli, Bosto, Greewood, RE., The umber of cycles associated with the elemets of a permutatio group, Amer. Math. Mothly, 6 (953) Heidel, J., Maloey, J., Farlow, C., ad Rogers, J., Fidig cycles i sychroous Boolea etwors with applicatios to biochemical sytems, It. J. Bif. ad Chaos, 3 (23) Kovalia, J., A uified iterpretatio of the biomial coefficiets, the Stirlig umbers, ad the Gaussia coefficiets, Amer. Math. Mothly, 7 (2), Mile-Thomso, L., The Calculus of Fiite Differeces, Chelsea, New Yor, 98. 5

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Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,