Forest volume decomp ositions and Ab el-Cayley-Hurwitz multinomial expansions Jim Pitman Technical Rep ort No. 591 Department of Statistics University of California 367 Evans Hall 3860 Berkeley,CA94720-3860 March 29, 2001; revised Octob er 24, 2001 Abstract This pap er presents a systematic approach to the discovery,inter- pretation and veri cation of various extensions of Hurwitz's multino- mial identities, involving p olynomials de ned bysumsover all subsets of a nite set. The identities are interpreted as decomp ositions of for- est volumes de ned bytheenumerator p olynomials of sets of ro oted lab eled forests. These decomp ositions involve the following basic for- est volume formula, which is a re nementofCayley's multinomial expansion: for R S the p olynomial enumerating out-degrees of ver- tices of ro oted forests lab eled by S whose set of ro ots is R, with edges directed away from the ro ots, is P P jS jjRj1 x : x s r s2S r 2R Supp orted in part by N.S.F. Grants DMS 97-03961 and DMS-0071448 1 1 Intro duction Hurwitz [21 ] discovered a numb er of remarkable identities of p olynomials in a nitenumber of variables x ;s 2 S ,involving sums of pro ducts over all s j j 2 subsets A of some xed set S ,witheach pro duct formed from the subset sums P x := x and x = x x , where A := A. 1 A s A A s2A For instance, for S =f0; 1g,withx substituted for x and y for x to 0 1 ease the notation, there are the following: Hurwitz identities [21, I I', I I I, IV] X jAj1 jAj j j xx + x y + x =x + y + x 2 A A A X jAj1 jAj1 j j1 xx + x y y + x =x + y x + y + x 3 A A A j j X X Y jAj jA j jB j x + x y + x = x + y + x jB j! x 4 A b A b2B A j j B j j For j j = n and x 1, by summing rst over A with jAj = k ,theseHurwitz s sums reduce to corresp onding Abel sums [1, 48 ] n X n ; k + nk + A x; y := x + k y + n k 5 n k k =0 for particular integers and . Esp ecially, 2 with x w and y = z nw s reduces to Abel's binomial theorem [1] n X n k 1 nk n xx + kw z kw =x + z k k =0 and 4 for x w reduces to a classical identity due to Cauchy. Strehl s [56] explains how Hurwitz was led to suchidentities via the combinatorial problem, which arose in the theory of Riemann surfaces [20 ], of counting the number of ways a given p ermutation can b e written as a pro duct of a minimal numb er of transp ositions which generate the full symmetric group. 2 This pap er develops a systematic approach to the discovery,interpre- tation and veri cation of identities of Hurwitz typ e such as 2-4 . This metho d is related to, but not the same, as Knuth's metho d in [32] and [31, Ex. 2.3.4.30]. There Hurwitz's binomial theorem 3 was proved byinter- preting b oth sides for p ositiveinteger x; y and x ;s 2 asanumber of rooted s forests with x + y + x + j j vertices, sub ject to constraints dep ending on the x . The present metho d is closer to the approachofFrancon[18], who de- s rived 2 and 3 using a form of Cayley's multinomial expansion over trees, expressed in terms of enumerator p olynomials for mappings from S to S . Section 2 intro duces the notion of forest volumes, de ned by the enumerator p olynomials of sets of ro oted lab eled forests, with emphasis on a general- ization of Cayley's multinomial expansion over trees, called here the forest volume formula. This formula relates Hurwitz typ e identities to decomp osi- tions of forest volumes. Probabilistic interpretations of forest volumes were developed in [40, 41, 42, 43]. This pap er, written mostly in combinatorial rather than probabilistic language, is a condensed version of [40 ]. Acom- bined version of [40] and [41 ] app ears in the companion pap er [44]. Section 3 o ers two di erent extensions of the forest volume formula. Then Section 4 presents a numb er of Hurwitz typ e identities, with pro ofs by forest volume decomp ositions. Finally, Section 5 p oints out some sp ecializations of these Hurwitz identities which give combinatorial interpretations of Ab el sums. 2 Forest volumes Ro oted lab eled forests. Francon's approach to Hurwitz identities is sim- pli ed byworking exclusively with various subsets of the set F := fall ro oted forests F lab eled by S g S whose enumerativecombinatorics has b een extensively studied [37 , 2.4 and 3.5], [54 , 5.3]. Each F 2 F is a directedgraph with vertex set S , that is a S subset of S S , each of whose comp onents is a tree with some ro ot r 2 S , with the convention in this pap er that the edges of F are directed away from the ro ots of its trees. The set of vertices of the tree comp onentof F ro oted at r is the set of all s 2 S such that there is a directedpath from r to s in F , F denoted r ; s, meaning either r = s or there is a sequence of one or more F F F F ! s means s ;s 2 F . Let F := fx : s ! xg edges r ! ! s, where s 2 1 2 s 1 3 denote the set of children of s in F . So jF j is the out-degree of vertex s s in the forest F . Note that the F are p ossibly empty disjoint sets with s [ F = S ro ots F , where ro ots F is the set of ro ot vertices F . So the s2S s P total numb er of edges of F is jF j = jS jjro ots F j and jro ots F j is s s2S the numb er of tree comp onents of F . Forest volumes. For B F ,theenumerator p olynomial in variables S x ;s 2 S s X Y jF j s V [F 2 B ]:=V [F 2 B ]x ;s 2 S := x 6 S S s s F 2B s2S is called here the volume of B , to emphasise that B ! V [F 2 B ]isa S measure on subsets B of F , for each xedchoice of x ;s 2 S withx 0. S s s As explained later in this section, this notion of forest volumes includes b oth the probabilistic interpretations develop ed in [43, 41 , 42 , 40 ], and the forest volume of a graph de ned by Kelmans [29, 26 , 27]. Consider the volume of all forests F 2 F with a given set of ro ots R. S This volume decomp oses according to the set A = [ F of children of r 2R r all ro ot vertices of F . With := S R,thisgives the recursivevolume decomp osition X jAj V [ro otsF =R]= x V [ro otsF =A] 7 R[ R A where each side is a p olynomial in variables x ;s 2 S ,and S = [ R with s R \ =;. Consideration of 7 for small jS j leads quickly to the following generalization of Cayley's multinomial expansion over trees [13, 47 ]: Theorem 1 The forest volume formula [13, 45, 47, 18, 11, 40] For R S , the volume of forests labeledbyS whose set of roots is R is jS jjRj1 V [ro otsF =R]=x x : 8 S R S Pro of. Observe rst that 8 transforms 7 into the following Hurwitz typ e identity of p olynomials in variables x and x ;s 2 : R s X j jjAj1 jAj j j1 x x : 9 x x + x = x A R R R A But 9 is very easily veri ed directly, and 8 follows from 7 and 9 by induction on jS j. 2 4 History and alternative pro ofs of the forest volume formula. Cayley [13] formulated the sp ecial case of 8 with R = fr g, call it the tree volume formula, along with the sp ecial case of 8 with general R and x 1;s 2 S , s that is the enumeration jS jjRj1 jfF 2 F with ro ots F =Rgj = jRjjS j 10 S n2 which for jRj = 1 yields the Borchardt-Cayley formula n for the number of unro oted trees lab eled by a set of n vertices. These sp ecial cases of the forest volume formula are among the b est known results in enumerativecom- binatorics. See for instance [45, 47 , 43 , 42, 54 , 7 ] for various pro ofs of the tree volume formula and [19, 36 , 37 , 57 , 51 , 3] for 10 . The preceding pro of of the forest volume formula parallels a pro of of 10 by induction, using the consequence of 7 that the numb er in 10 is jS j; jRj with the recursion n X n a k + n; k = k n; a: 11 a a=1 Mo on [37 , p. 33] attributes this pro of of 10 to Gob el [19]. The forest volume formula can also b e derived by the metho d of Prufer co des [45 ], which has b een applied to obtain a host of other results in the same vein [30],[37, Chapter 2].