An Empirical Method for Solving (rigorously!) Algebraic Functional Equations Of the Form F(P(x,t), P(x,1),x,t)=0
aa r X i v : . [ m a t h . C O ] D ec An Empirical Method for Solving (Rigorously!) Algebraic-Functional Equations of the FormF ( P ( x , t ) , P ( x , ) , x , t ) ≡ By Ira M. GESSEL and Doron ZEILBERGER
Preface: It is Never Too Late
One of the references in Doron Zeilberger’s article [Z1], written more than 23 years ago, was to anarticle, labeled in planning , with the authors and title of the present article. We must have bothforgotten our commitment. But better late than never , so with a delay of almost a quarter-century,at long last we got around to writing the promised article. But it is just as well that we waited,since now computers are much faster, and we are much better programmers!
Functional Equations
Many enumeration problems reduce to solving a functional equation for a generating function withrespect to one or more variables that we do care about, and one or more variables, that we don’t care about, called catalytic variables . At the end of the day we set all the catalytic variables to 1(or sometimes 0, and possibly other specific numbers). Even though—at least initially—we donot care about these extra variables (corresponding to auxiliary combinatorial statistics), they areneeded to set up the combinatorial reaction , so to speak, only to be discarded, like their chemicalnamesakes, once the ‘reaction’ is finished.Initially humans solved these, one at a time, using ad hoc human ingenuity, and this goes back tothe pioneering work of Tutte and his school on counting maps, in the early 1960s.More recently, the favorite method became the powerful and sophisticated kernel method that hadscored many triumphs in the hands of such virtuosi as CNRS Silver-Medalist Mireille Bousquet-M´elou and other people. For lucid and engaging overviews, see the slides of the talks [B1] and [B2].Alas, this method, in addition to requiring a lot of human ingenuity , is also very human-labor-intensive .It turns out, that in many cases (perhaps all!), there is an alternative, much simpler, approach,based on empirical guessing , yet it is fully rigorous! Of course, this method requires the help ofour silicon colleagues.
The Zeilberger Gordian Knot
In Zeilberger’s proof [Z1] of Julian West’s [W] conjectured explicit expression for the number of2-stack-sortable permutations, it was necessary to solve the functional equation f ( x, t ) = 11 − xt + xt ( f ( x, − tf ( x, t ))( f ( x, − f ( x, t ))(1 − t ) . (FunEq)By clearing denominators, this equation can be written more compactly as F ( f ( x, t ) , f ( x, , x , t ) ≡ , F of four variables.The idea is extremely simple. Since f ( x,
1) was conjectured to be a certain known algebraic formalpower series, why not guess that this is also the case for the two-variate formal power series f ( x, t );i.e., let the computer guess a polynomial in three variables—let’s call it G —such that G ( f ( x, t ) , x , t ) ≡ . (AlgEq)Once guessed, it is purely routine to prove our guess rigorously. Since both (FunEq) and (AlgEq)have unique formal power series solutions, after we define g ( x, t ) to be the unique solution of(AlgEq), the verification that F ( g ( x, t ) , g ( x, , x , t ) ≡ , is a routine calculation in the Sch¨utzenberger ansatz [Z2]. In fact, since we know a priori that theleft side satisfies some algebraic equation, all we need is to bound the degrees, and check that thefirst few coefficients (in x ) are identically 0. Since that’s how we got it in the first place, we alreadyknow that! Nevertheless, the pedantic purist may want to bound the degrees exactly. The Gessel ShortcutAlas , guessing the three-variate polynomial G takes a very long time. After the first draft of [Z1]was written, Ira Gessel made the following observation, reproduced at the very end of the finalversion of [Z1], that we now reproduce.“ Epilogue: How The Proof Could Have Been Found
July 2, 1991: The first proof of any conjecture is seldom the shortest. It turns out that the presentproof is no exception. Ira Gessel made the brilliant observation that steps 2–5 can be replaced bythe following.
Step ′ : Conjecture a polynomial I , of two variables, such that I ( P ( x ) , x )) = 0 where P ( x ) = f ( x, f ( x, t ) that satisfies F ( f ( x, t ) , f ( x, , x, t ) ≡ I ( f ( x, , x ) ≡
0. Let’s write,(i) F ( f ( x, t ) , Q ( x ) , x, t ) ≡ , (ii) f ( x,
1) = Q ( x ) , (iii) I ( Q ( x ) , x ) ≡ . We have to prove that (i)+(ii) implies (iii). But note that (i)+(ii) have a unique solution, and(i)+(iii) have a unique solution, and we must show that these are the same. So it’s enough to showthat (i)+(iii) implies (ii). Taking the resultant of F and I w.r.t. Q ( x ) gives the algebraic equation G ( f, x, t ) ≡ Q ( x ) = P ( x ). This observation is the leitmotif of a paper that Ira Gessel and I hope towrite.” 2 mplementation This method turns out to be applicable to many other functional equations. Using Maple, it iseasy to follow Gessel’s advice. It is very fast to guess a polynomial of two variables, let’s call it I ,such that I ( f ( x, , x ) ≡ . By the assumption, we have the relation F ( f ( x, t ) , f ( x, , x , t ) ≡ . Now eliminate f ( x,
1) from both equations, using, say, Maple’s built-in command resultant , andget an algebraic equation linking f ( x, t ), x , and t , without f ( x, f ( x, t ) by f ( x, t by 1, and get a polynomial in f ( x,
1) and x and make sure that it is a nonzero multiple of I .This is so much faster than the original approach, and all the steps are fully automatic. Linear Recurrence Equations with Polynomial Coefficients
It is well-known, and easy to see (and implement, e.g., in the Maple package gfun described in [SZ])that once a formal power series satisfies an algebraic equation, as above, it also satisfies a lineardifferential equation with polynomial coefficients (i.e., is D -finite), and hence the enumerating se-quence itself, satisfies a linear recurrence equation with polynomial coefficients (i.e., is P -recursive).All this can be found automatically, and in fact, since we know a priori that such a recurrenceis bound to exist, it is completely legitimate to guess it empirically. If we are lucky, and thatrecurrence happens to be first-order , then we get a closed-form ‘elegant’ expression, like in the caseof [W], first proved in [Z1]. The Maple Package FunEq
Everything here is fully implemented in the Maple package
FunEq , available from the front of thisarticle: . It alsocontains quite a few sample input and output files that readers are welcome to extend. In particular,we give fully rigorous automatic proofs of all the results in [CJS] (and many other ones, where theanswer is not ‘nice’), as well as a two-second proof of the main result of [Z1], and we solve numerousother functional equations that we picked more or less at random, just to test the method.
Future Work: More Catalytic Variables; Beyond the Sch¨utzenberger Ansatz
The present, naive, guessing approach should be applicable to functional equations with more thanone catalytic variable, but then, according to [B1] and [B2], one may have to go to other ansatzes,first D -finite, and then formal power series that satisfy an algebraic differential equation ratherthan a linear one. Sooner or later, things would become too difficult even for computers, but onecan do lots of shortcut tricks, and we are sure that the present empirical-yet-rigorous approachshould be extendible, and implementable . Whether you would get nice results that humans canenjoy remains to be seen, and is rather unlikely. Hence it may not be worth the effort.3 eferences [B1] M. Bousquet-M´elou, Enumeration with “catalytic” parameters: a survey , Three lectures at the67th S´eminaire Lotharingien de Combinatoire, Bertinoro, Italy, September 2011. Available from .[B2] M. Bousquet-M´elou,
M´eli-m´elo(u) de combinatoire , expos´e pour la remise de la m´edailled’argent du CNRS, Bordeaux, Oct. 2, 2014. Available from .[CJS] R. Cori, B. Jacquard, and G. Schaeffer,
Description trees for some families of planar maps ,“Proceedings of the 9th Conference on Formal Power Series and Algebraic Combinatorics ” (Vienna,1997), 196–208. Available from .[SZ] B. Salvy and P. Zimmermann,
GFUN: a Maple package for the manipulation of generatingand holonomic functions in one variable , ACM Trans. Math. Software (1994), 163–177.[W] J. West, Sorting twice through a stack , Theoretical Computer Science (1993), 303–313.[Z1] D. Zeilberger,