The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ... , n are D-finite for all d and all r
aa r X i v : . [ m a t h . C O ] D ec The Generating Functions Enumerating 12 ... d-Avoiding Words with r occurrences of eachof 1 , . . . , n are D-finite for all d and all r
By Shalosh B. EKHAD and Doron ZEILBERGERDedicated to Neil James Alexander Sloane (born October 10, 1939), on hisA000027[75]-th birthday (alias A005408[38]-th, alias A002808[53]-th, alias A001477[74]-th, aliasA014612[17]-th, and 13520 other aliases.)
Introduction
In a recent beautiful article, Nathaniel Shar and Doron Zeilberger ([ShZ]) proved that for anypositive integer r , the generating function of the sequence enumerating 123-avoiding words with r occurrences of each of the letters 1 , . . . , n is always algebraic . In other words for each r , thegenerating function, let’s call it f r ( x ), satisfies an equation of the form P r ( x, f r ( x )) = 0 , for some polynomial, P r , of two variables. The actual polynomials, P r ( x, y ), were computed for r ≤ . . . d -avoiding words with d ≥ even for r = 1.In 1990 Doron Zeilberger ([Z]) showed that for and any positive integer d , the generating functionenumerating 1 . . . d -avoiding permutations (i.e. words in { , . . . , n } where each letter occurs exactly1 times) is the next-best-thing to being algebraic, which is being D-finite (aka as holonomic ). Recallthat a formal power series is D -finite if it satisfies a linear differential equation with polynomialcoefficients, or equivalently, the enumerating sequence itself is P-recursive , i.e. satisfies a linear recurrence equation with polynomial coefficients. Ira Gessel ([G]) famously discovered (and proved)a beautiful determinant with Bessel functions, for the generating function, (of the sequence dividedby n ! ) (that also implies the above result), and Amitai Regev ([R]) famously derived delicate andprecise asymptotics.In the present article, dedicated to guru Neil Sloane on his 75-th birthday, we observe that theanalogous generating functions for multi-set permutations (alias words ), where every letter appearsthe same number of times, say r , are still always D-finite , (for every d and every r ), and we actuallycrank out the first few terms of quite a few of them, many of whom are not yet in the OEIS ([Sl]).All this data, often with linear recurrences (that we know a priori exist, and hence it justifies theirdiscovery by pure guessing), and very precise asymptotics, is collected in the front of this article SLOANE75 and
NEIL , can be found and downloaded, and readers who have Maple and computer time to spare arewelcome to use in order to generate yet more data.Last but not least, we pledge 100 dollars to the OEIS in honor of the first one to prove our con-jectured asymptotic formula for the number of 1 . . . d -avoiding words in { r . . . n r } that generalizesRegev’s ([R]) famous formula for r = 1. We pledge another 100 dollars for extending Gessel’s Besseldeterminant, from the r = 1 case to general r . Why is the Sequence Enumerating 1 . . . d-avoiding words in { r . . . n r } P-recursive?
By the Robinson-Schenstead-Knuth (RSK) famous correspondence, our quantity of interest, let’scall it A d,r ( n ) is given by A d,r ( n ) = X λ ⊢ rnlength ( λ ) ≤ d f λ g ( r ) λ , where f λ is the number of standard Young tableaux of shape λ = ( λ , . . . , λ d ), and g ( r ) λ is the numberof column-strict Young tableaux with exactly r occurrences of each of 1 , . . . , n . For λ = ( λ , . . . , λ d )(where we pad it with zeroes if the length is less than d ), f λ is closed-form (thanks to Young-Frobenius, or the hook-length formula), and hence ipso facto , holonomic in its d discrete arguments.Furthermore, for r > g ( r ) λ , while no longer closed-form, is easily seen to be holonomic in its d discrete arguments (one way to see this is to note that their redundant generating function (in thesense of MacMahon) is a rational formal power series in x , . . . , x d ). It follows, by general holonomicnonsense ([Z]), that for any fixed integers r and d the sequence, in n , { A d,r ( n ) } , is P -recursive.Computationally speaking, it is fairly easy to compute g ( r ) λ , and hence crank-out the first few termsof the sequences { A d,r ( n ) } for quite a few d and r , that for r and d not too large may be used toguess (in real time) the recurrences empirically, that we know must be the right ones. The 100 dollars conjecture generalizing Regev’s AsymptoticsConjecture (100 donation to the OEIS in honor of the first prover)Let A d,r ( n ) be the number of 1 . . . d -avoiding words in { r . . . n r } , then there exists a constant C r,d such that A d,r ( n ) ∼ C r,d · (cid:18)(cid:18) d + r − d − (cid:19) ( d − r (cid:19) n · n (( d − − / . Extra Credit (25 additional dollars): find an explicit expression for C r,d in terms of r and d (involving π , of course). The 100 dollars Challenge to generalize Gessel’s Spectacular Theorem
This is more open-ended, but it would be nice to get a determinant expression, in the style of IraGessel’s ([G]) famous expression for the generating function of A d, ( n ) /n ! , canonized in the bible u k ( n ) := A k, ( n ), then X n ≥ u k ( n ) n ! x n = det( I | i − j | (2 x )) i,j =1 ,...,k , in which I ν ( t ) is (the modified Bessel function) I ν ( t ) = ∞ X j =0 ( t ) j + ν j !( j + ν )! . Guru Herb Wilf (ibid) goes on to wax eloquently: “ At any rate, it seems fairly “spectacular” to me that when you place various infinite seriessuch as the above into a k × k determinant, and then expand the determinant, you should find thatthe coefficient of x n , when multiplied by n ! , is exactly the number of permutations of n letterswith no increasing subsequence longer than k .” It would be even more spectacular, if you, dear reader, would generalize this to r > References [G] I. Gessel,
Symmetric functions and P-recursiveness , Journal of Combinatorial Theory, Series A (1990), 257-285; http://people.brandeis.edu/~gessel/homepage/papers/dfin.pdf .[R] A. Regev, Asymptotic values for degrees associated with strips of Young diagrams , Adv. Math. (1981), 115-136.[ShZ] N. Shar and D. Zeilberger, The (ordinary) generating functions enumerating -avoidingwords with r occurrences of each of , , ..., n are always algebraic , submitted; .[Sl] N. J.A. Sloane, The Online-Encyclopedia of Integer Sequences (OEIS) https://oeis.org/ .[W] H. Wilf, Mathematics, an experimental science , in: “Princeton Companion to Mathematics”,(W. Timothy Gowers, ed.), Princeton University Press, 2008, 991-1000; .[Z] D. Zeilberger,