Proof of Geeratig Fuctio For J.B.S.A.R.D.T.a.a.

Transcription

1 Ca. J. Math., Vol. XXXVII, No. 6, 1985, pp DIRECTED GRAPHS AND THE JACOBI-TRUDI IDENTITY I. P. GOULDEN 1. Itroductio. Let \a i L X deote the X determiat with (/', y)-etry a-, ad h k = h k (x x,...,x ) deote the k l -homogeeous symmetric fuctio of x x,..., x defied by A* = 2*7"...;C" where the summatio is over all m v..., m ^ 0 such that m x -f... + m = k. We adopt the covetio that h k = 0 for k < 0. For itegers a x ^ a 2 =... = a ^ 0, the Jacobi-Trudi idetity (see [6], [7] ) states that ^^-i+frix = \ x i J \X/\ x i \X' I this paper we give a combiatorial proof of a equivalet idetity, Theorem 1.1, obtaied by movig the deomiator o the RHS to the umerator o the LHS. THEOREM 1.1. For a, ^... ^ a i^ 0, we have \ X i \X\ a l -i+j\x = \ X i J \X- Our proof is obtaied by adoptig the followig strategy for provig a idetity, say f = g. We defie a set 9) of combiatorial objects ad a weight fuctio for the elemets of Q), such that the geeratig fuctio for 2 with respect to this weight is / (with o cacellatio). We fid a subset srf of 3) for which the geeratig fuctio is g (agai, o cacellatio of terms). We the fid a ivolutio o Q) stf such that the weight of each elemet of 2 stf is equal to the egative of the weight of its image uder the ivolutio (so we say the ivolutio is weight-reversig). This immediately proves that the geeratig fuctio for 3) stf is equal to 0, ad so the idetity follows. By cosiderig touramets as the combiatorial objects, this "ivolutioal" method has bee applied by Gessel [3] to prove the Vadermode determiat formula. Bressoud [1] has geeralized this to prove the Weyl deomiator formulae for the root systems B, C ad D. I this case the combiatorial objects are touramets i which the edges are coloured ad the vertices are "hadicapped". A ivolutio for «-tuples of Received Jue 27, This work was supported by a grat from the Natural Scieces ad Egieerig Research Coucil of Caada, ad was carried out while the author was visitig the Departmet of Mathematics, M.I.T. 1201

2 1202 I. P. GOULDEN lattice paths is used by Gessel ad Vieot [4] (see also [5], Sectio 5.4) to prove that the LHS of the Jacobi-Trudi idetity equals the geeratig fuctio for colum-strict plae partitios with shape (a ]9..., a ). I the preset paper this ivolutioary method is applied to a set of directed graphs. All of the above applicatios of the ivolutioary method give elegat ad isightful proofs of results for which algebraic proofs are kow. It is worth metioig that Zeilberger ad Bressoud [8] (see also [2] for a geeralizatio) have obtaied a ivolutioary proof of the #-Dyso Theorem, for which o other proof is kow. I Sectio 2 of this paper we defie a set of directed graphs 3, a weight fuctio wt, ad a subset stf of 3 such that the geeratig fuctios for 3 ad J3^ respectively, are the LHS ad RHS of Theorem 1.1. We also partitio 3 stf ito two sets, 38 ad #. Sectio 3 gives weight-reversig ivolutios for SS ad ^separately, fiishig a proof of Theorem 1.1 by the ivolutioary method. Fially, we defie some terms. A touramet o vertices is a loopless directed graph with labelled vertices l,...,w i which every pair of vertices (ij) is joied by exactly oe edge, either directed from / toy, or from y to /'. A trasitive touramet is a touramet with o directed cycles. For ay trasitive touramet there exists a permutatio a e S such that vertex o(i) has out-degree /, for i = 1,...,. We say that o is the wier permutatio of the trasitive touramet, ad p = o()o( 1)... a(l) is the loser permutatio (this is motivated by supposig that a edge directed from i toy meas that perso / "beats" perso j). We deote the sig of the permutatio o by sg(a). 2. Directed graphs. Cosider directed loopless graphs o I labelled vertices, which cosist of "black" vertices, labelled l,...,w, ad "white" vertices, labelled 1,...,. For fixed a = (a b..., a ), where «j ^ a 2 =... = a i^ 0, let 3 cosist of all such graphs i which: (i) the edges icidet with black vertices oly form a trasitive touramet o vertices (called the black touramet). (ii) the edges icidet with white vertices oly form a trasitive touramet o vertices (called the white touramet). (iii) the edges icidet with both a black ad a white vertex are all directed from the black vertex to the white vertex, with multiple edges allowed. (iv) the oly restrictio is that i-degree (white vertex /) = a t + i, for / =!,...,«. For D G 3, let a t = out-degree (black vertex /), for / = 1,...,«, let a be the wier permutatio of D's black touramet, ad let p be the loser

3 DIRECTED GRAPHS 1203 permutatio of D's white touramet. Defie a weight for D, deoted wt(z)), by wt(d) = sg(a) sg(p) JC?»... x a \ ad for ay subsets c ^ let the geeratig fuctio for y with respect to this weight be deoted by 0(^), so 4>(^) = 2 wt(z>). Fially, let N(i,j) deote the umber of edges directed from black vertex / to white vertex j i D, where Uj= 1,...,. For example, for a = (4, 2, 2, 0) the graph E x i Figure 1 is i 2, with a = 3241, p = 2134, sg(a) = 1, sg(p) = -1, N(2, 1) = N(3, 1) = 2, JV(1, 2) = JV(2, 3) = N(4, 1) = N(4, 3) = 1, ad wt(,) = X ixi^i^i. > Jf > «y> < Figure 1 : A elemet of Q for a = (4, 2, 2, 0). Note that E x is draw with the vertices arraged i two rows. The top row cotais the black vertices, i order a(l),..., o() from left to right ad the bottom row cotais the white vertices, i order 1,..., from left to right. Moreover, white vertex / is directly below black vertex a(/), for /' = 1,...,. We shall follow this covetio whe drawig elemets sice it allows a coveiet geometrical descriptio of the bijectios i Sectio 3. Coditio (iv) o 2 implies that if OLJ = 0, the p(y) = j ad N(o(i),j) = 0 for / = 1,...,. This is illustrated by E v where a 4 = 0. The sigificace of the set 3) is revealed by the ext result, which shows that the LHS of Theorem 1.1 is the geeratig fuctio for 2.

4 1204 I. P. GOULDEN PROPOSITION 2.1. For a x =" a 2 =... = a i^ 0, we have /V00/! Let i? cosist of all elemets of 3) whose black touramet has wier permutatio a ad whose white touramet has loser permutatio p. The clearly *( ) = 2 2 *(\ P ). a^s p^s Now from the defiitio of wier ad loser permutatio, for elemets of S) ap we have "o(k) = - k+ % N(a(k) 9 p(l) ), k = 1,..., /!, ad restrictio (iv) for <@ yields - I + 2 JV(a(*), p(/)) = a M + «- p(/), k = 1 or equivaletly, (*) 2 JV(0(*), p(/) ) = Up(l) - p(/) + /, / = 1,...,. The defiitio of 0 ow gives *(0 o, p ) = sg(a) sg(p) 2 Û x "- k + & N( (kmi) \ /c= 1 where the summatio is over N(o(k), p(l) ) i^ 0 for k, I = 1,...,, subject to restrictio (*). Thus *(^p) = sg(a) sg(p) II x^f II A, where A^s*^*'» i which the summatio is over all N(o(k), p(l) ) = 0 for k = 1,..., w, with the restrictio that 2 tf(a(*), P(/) ) = «*/) " P(/) + / A:=l

5 DIRECTED GRAPHS 1205 From the defiitio of the homogeeous symmetric fuctios of variables, it follows that so *(#) = 2 sg(a) It *J f 2 sg(p) ]I h {l)+h m oes k=\ pes /=1 ad the result follows immediately. Now let s/ be the subset of S cosistig of all graphs A m S m which: (i) the white touramet of A has loser permutatio p = (ii) N(o(i), j) = 0 for / = j, i, j = 1,..., where a is the wier permutatio of yts black touramet. For example, if a = (4, 2, 2, 0) the the graph E 2 i Figure 2 is i jtf, with a = 3241, ad wt(2s 2 ) Note that = x i- x 2- x 3- x 4- Figure 2: A elemet of s/ for a = (4, 2, 2, 0). coditio (i) forces all edges betwee white vertices to be directed from right to left, ad coditio (ii) forces ay edge from a black vertex to a white vertex to lie i a sigle colum. The sigificace of the subset sf of S is revealed by the ext result, which shows that the RHS of Theorem 1.1 is the geeratig fuctio forj^

6 1206 I. P. GOULDEN PROPOSITION 2.2. For a, ^ a 2 =... = a â 0, we have <H*) = \x? + -\ x. Proof. Let si a cosist of all elemets of si whose black touramet has wier permutatio a. The clearly aes For elemets of J^ we have a o(k) = - k + #(a(*), *) for /c = 1,..., by coditios (i) ad (ii) for si. Also, restrictio (iv) for 3) applied to elemets of si 0 yields so a k + rl-l c = -l c + N(a(k), k) for k = 1,...,, N(o(k),k) = <x k for A: = 1 /i. Combiig these results, we fid that si a cosists of a sigle graph, for which a a ( k ) = a k + k, k = 1,...,, so *C*0 = 2 sg(a) f[ JC^""*, ags k=\ ad the result follows immediately. Let <% be the subset of 3) cosistig of those graphs i which N(o(i),j) > 0 for some w ^ i > j ^ 1. Clearly si ad SI are disjoit, sice the existece of such i > j violates coditio (ii) for si. Fially, we let V = 2) - si - ai, so Si ad ^partitio 3) - si. I Sectio 3 we complete a ivolutioary proof of Theorem 1.1 by fidig weight-reversig ivolutios for Si ad #. 3. Ivolutios ad the Jacobi-Trudi idetity. First we cosider a mappig \p for Si. For D e 3 let #(/)) = { (i,j) \N(o(i),j) >0,^i>j^\}. Now Si cosists precisely of those graphs B for which S?(B) = 0. For example, E x i Figure 1 is i Si, sice #(,) = { (2, 1), (3, 1), (4, 2) } * 0. Thus, for 5 e #, if (J, 0 = Max(Mi{<^(5) } ) ' J the (s, /) is well-defied. We obtai \j;(b) = B' from B as follows: reverse

7 DIRECTED GRAPHS 1207 the edge (o(s 1), o(s) ) i the black touramet, ad replace oe of the N(o(s), t) > 0 edges from black vertex o(s) to white vertex t by a additioal edge from black vertex o(s 1) to white vertex /. For example ^(E x ) = E 3 ad ^(E 3 ) = E h where E ]9 E 3 G SS are give i Figures 1 ad 3. Note that wt( 3 ) = x\x\x\x\= -wt(,). Figure 3: A image uder $ for a = (4, 2, 2, 0). We ow prove that ^ is a weight-reversig ivolutio i geeral. THEOREM 3.1. The mappig \f/:<% I > B f is a ivolutio with wt(^) = -wt( ), so <&(&) = 0. Proof. Let the parameters of the graph B' be deoted by a prime (e.g., the loser permutatio of the white trasitive touramet is deoted by p' ad the out-degree of black vertex / by a[). Sice o edges i the white touramet are affected, we have p' = p. Sice o(s 1) ad a(s) are cosecutive i the wier permutatio of the white trasitive touramet, reversig the edge betwee them yields a trasitive touramet, with o(s 1) ad o(s) tradig places i the wier permutatio, so a' = a(l)... o(s - 2)o(s)o(s - l)o(s + 1)... a(). The out-degrees of black vertices a( 1 ),..., o(s 2), o(s + 1),..., o() are uaffected by i//, ad the black vertices a(s 1) ad a(s) each has oe out-directed edge replaced by aother out-directed edge, so a\ = a t for / = 1,...,. Similarly, the i-degrees of the white vertices are uchaged by;//.

8 1208 I. P. GOULDEN Sice the oly edges betwee black ad white vertices that are chaged are those betwee black vertices o(s 1) = o'(s), o(s) = o'(s 1) ad white vertex t, we obtai N'(o'(i)J) = N(o(i) 9 j), ij = 1,...,*; / * s - 1, 5, #'(*'(* - \),j) = N(o(s) 9 j) J = 1 «;y * r, N'(o'(s)J) = N(a(s - \)J)J = 1,...,*;./# f, #'(*'(* - 1), 0 = #(*(*), 0, JV'(a'(s), 0 = N(a(s - 1), Thus we have show that B' e ^, sice (s, t) e ^(1?') ad coditios /**\ (i)-(iv) for 3) are satisfied. Also wt(i?') = wt(i?), sice sg(a') = sg(a). We complete the proof of this result by showig that B" = B, where B" = ^(B'). A ispectio of (**) shows that {j\ (ij) for some /' = 1,..., ) = {j\ 0J) ^ &(B') for some i = 1,..., w}, so / = t\ sice / ad f are the smallest elemets of these two sets, respectively. Similarly {/ (/, t) G (B) } U {s - 1} = {/ (/, /) e (B') } U {s - 1}, ad $' = s. Thus (V, /') = (s, t), so a" = a, p" = p, ad two applicatios of (**) yields N"(o'\i)J) = N(o(i),j), for ij = 1,..., *. Thus i?" = B ad the result follows. Now we cosider a mappig for C. If D G j^ â?, let ^ (D) = { (1,7) tf (a(i), po) ) > 0, 1 ^ i < j ^ ). For example, for E 4, E 5 (<E <g) i Figure 4, J^( 4 ) = { (1, 2), (1, 3), (2, 3) }, JF( 5 ) = { (1, 3) }, ad for 2 (e sf) i Figure 2, ^(E 2 ) = 0.

9 DIRECTED GRAPHS 1209 Figure 4: Two elemets of #for a = (4, 2, 2, 0). I fact, it is true i geeral that for D e 2 - ^^{D) = 0 if ad oly if D e i The "if" part of this statemet is true sice D e i implies that p(j) = J f r au, y> so N( a (0> P(j) ) = 0 for all i ^ y, ad thus / < y*. This, ad the fact that D & also tells us that the "oly if part is true for p = To prove the "oly if part for other p, assume JF(D) = 0 for D s/, ad let k be the maximum y such that p(j) * y. The k ^ 2, p(fc) < A: ad N(o(i),j) = 0 for /' ^ y whe /' > A: or y > k, sice J*"(Z)) = 0 implies N(a{i)MJ)) = 0 for/ < y, ad D 38 implies N(a(i)J) = 0 for/ > y. These combie to give iv(a(/), p( ) ) = 0 But the i-degree restrictio gives for all / = 1,..., AI. 2 N(o(i), p(k)) = a m - p(k) + k > 0, /= 1 sice a^k) proof. ^ 0 ad k > p(&), so we have a cotradictio, fiishig the Thus for C e «; if (w, v) = Max(Mi{J^(C) }, j i the (w, v) is well-defied. We obtai (C) = C from C as follows: reverse the edge (p(v), p(v 1) ) i the white touramet, ad replace oe of the N(o(u), p(v)) edges from black vertex o{u) to white vertex p(v) by a

10 1210 I. P. GOULDEN additioal edge from black vertex o(u) to white vertex p(v 1). For example ( 4 ) = E 5 ad (E 5 ) = E 4. Note that wt( 4 ) = x } x 2 x 3 x 4 = wt(e 5 ) ad (w, v) = (1, 3) for both graphs, which holds i geeral by the ext result, completig the ivolutioary proof of Theorem 1.1. THEOREM 3.2. The mappig :# > #:C H-> C is a ivolutio with wt(c') = wt(c), so ${< ) = 0. Proof. We follow closely the proof of Theorem 3.1, sice \p ad are very similar, ad omit details that are commo. The out-degrees of black vertices, i-degrees of white vertices, ad the permutatio a, are all uaffected by. The positios of p(v 1) ad p(v) are iterchaged i p to create p' with sg(p') = sg(p), so for all C G % we have created C e 3) with wt(c') = wt(c). To prove that is a ivolutio (so C e #) it is sufficiet to prove that (w', v') = (w, v). Now the miimality of u implies N(a(i)MJ)) = 0 for/ < j, i < u. We obtai p(/') = / for / = 1,..., u 1 ad N(o(i),j) = 0 for i < u or j < u by the same argumet (reversed) that was used above to prove that &(D) = 0 implies Z)e jaf for D G ^ - SI. Thus p(v - 1) ^ w, so C 96. Furthermore, (w, v) G J^Z)') while (i,j) ^(D f ) for / < w, so w r = u ad C s/, givig C G #. Fially, the maximality of v implies that there are o edges betwee black vertex o(u) ad white vertex p(y) = p'(y) for j > v i both C ad C, so v r = v. REFERENCES 1. D. M. Bressoud, Colored touramets ad Weyl's deomiator formula, preprit. 2. D. M. Bressoud ad I. P. Goulde, Costat term idetities extedig the q-dyso theorem, Tras. Amer. Math. Soc. (to appear). 3. I. Gessel, Touramets ad Vadermode's determiat, J. Graph Theory 3 (1979), I. Gessel ad G. Vieot, Determiats ad plae partitio, preprit. 5. I. P. Goulde ad D. M. Jackso, Combiatorial eumeratio (J. Wiley, New York, 1983). 6. I. G. Macdoald, Symmetric fuctios ad Hall polyomials (Claredo Press, Oxford, 1979). 7. R. P. Staley, Theory ad applicatios of plae partitios: Parts I ad II, Studies i applied mathematics 50(1971), ; D. Zeilberger ad D. M. Bressoud, A proof of Adrews' q-dyso cojecture, Discrete Math. 54(1985), Uiversity of Waterloo, Waterloo, Otario