A Set of Conjectured Identities for Stirling Numbers of the First Kind
aa r X i v : . [ m a t h . C O ] A ug A Set of Conjectured Identities for Stirling Numbers of theFirst Kind
Paul FederbushDepartment of MathematicsUniversity of MichiganAnn Arbor, MI, 48109-1043August 29, 2018
Abstract
Given an integer g , g ≥
2, an integer w , 0 ≤ w ≤ g −
2, and a set of g distinct numbers, c , ..., c g , we present a conjectured identity for Stirling numbers of the first kind. We haveproven all the equalities in case g ≤
6; and for the case g = 7, provided w ≤
3. Theseexpressions arise from an aspect of the study of the dimer-monomer problem on regulargraphs.
We organize the paper into two parts. In the first part we present the conjectured identities .In the second part we sketch their origins in the monomer-dimer problem, arising from work ofM. Pernici, [2]. A brief discussion of proofs is given at the end of Part 1.Part 1 Conjectured IdentitiesThe (unsigned) Stirling numbers of the first kind, (cid:20) ab (cid:21) , are defined by x ( x + 1) ... ( x + n −
1) = n X k =0 (cid:20) nk (cid:21) x k (1)See [1]. It is easy to show (cid:20) nn − w (cid:21) is a polynomial in n of degree 2w. So we may naturally define (cid:20) xx − w (cid:21) for any number x by extending the domain of the polynomial. We set P w ( x ) ≡ (cid:20) xx − w (cid:21) (2)Now we give ourself an integer g ≥
2, an integer w , 0 ≤ w ≤ g −
2, and a set of g distinctnumbers, S = { c , ..., c g } (3)We define a configuration as a sequence of non-empty subsets of S S , S , ..., S r (4)1hat are disjoint with union S , i.e. S i = ∅ , S i ∩ S j = ∅ if i = j, r [ i =1 S i = S (5)For a configuration we define t i = X c i ∈ S i c i , i = 1 , ..., r (6)A weighted configuration is a configuration as above for which each S i is assigned a non-negativeinteger, w i , its weight, with the restriction r X i =1 w i = w (7)Such a weighted configuration has an evaluation defined as( − r r Π i P w i ( t i ) (8)The conjectured identity is that the sum over all distinct weighted configurations of their evalu-ations is zero.We have proved the identities in certain cases by directly computing the sum of evaluations.In this way we verified the identities for g ≤ g ≤ g = 7 and w ≤
3. The computer computations weredone on a desktop computer using Maple in a couple of days. Clearly one could go further bycomputer. We have no idea how to organize a general proof (if one exists), we expect any generalproof to be extremely difficult. It is possible that someone may find inspiration to a line of prooffrom the source of the identities sketched in Part 2.Part 2 Through the Monomer-Dimer Problem, More General Conjectured IdentitiesI. Wanless developed a formalism for the monomer-dimer problem on regular graphs that isthe source of all our conjectured identities, [3]. More particularly we take our formulas from thework of M. Pernici [2], a systemization of Wanless’s results. We extract from this treatment ofthe monomer-dimer problem only the formulas that directly lead to our identities. We feel thisis the only material that might stimulate ideas towards a proof.We use three equations, (10), (12), and (16) from [2], with slight modification: M j ( n, r ) = [ x j ]exp( nrx − X s =2 nu s ( r ) s ( − x ) s ) (9) M j = n j r j j ! j − X h =0 a h ( r, j ) n h (10)[ j k n − h ]ln(1 + j − X s =1 a s ( r, j ) n s ) = 0 , k ≥ h + 2 (11)The symbol [ x j ] extracts from the expression following it the coefficient of the x j term in itsexpansion in powers of x. Likewise the symbol [ j k n − h ] extracts from the expression following itthe coefficient of the j k n − h term in an expansion in powers of j and n .2n [2] Pernici with a computation using an idealized physical argument derives equation (11)from equations (9) and (10) for particular values of the u s ( r ), s ≥
2. This result is checked in[2] for a large number of cases. We now state a set of conjectured identities more general thanthose in Part 1.For any set of values u s ( r ) , s ≥ g, w , c , ..., c g where the c i are distinct integers ≥
2. Uponsome reflection it is easy to see that verifying the identities of this type suffices to verify all theidentities of Part 1.We set k = X c i − w (12)and h = X c i − g (13)so that k − h = g − w ≥
2. And we set all the u s to zero except for values of s equal to oneof the c i . Then the left side of (11) is identically equal to zero as a function of such u s . Welook at the coefficient of the term u c u c ...u c g in a power series expansion of the left side of (11)in the u s . The statement that this coefficient is zero is the desired conjectured identity of Part1. It is perhaps not trivial to verify this. ( We think it should not be too difficult to prove theimplication in the other order, that the identities of Part 1 imply the identities of Part 2. )Part 3 PostscriptI find the three sets of conjectured identities (those of Part 1, those of Part 2, and those of[4]) very mysterious. A colleague of mine has said of one of the sets that he thinks once a proofis found it will be easy. My feeling is the opposite, that they are candidates for being things truebut not provable.ACKNOWLEDGMENT I would like to thank John Stembridge for much useful information. References [1] https://en.wikipedia.org/wiki/Stirling numbers of the first kind[2] Pernici, M., n Expansion for the Number of Matchings on Regulars Graphs and Monomer-Dimer Entropy , J. Stat. Phys. (2017) 168: 666–679.[3] Wanless, I. M.,
Counting Matchings and Tree-like Walks in Regular Graphs , Combin. Probab.Comput. 19, 463–480 (2010)[4] Federbush, P.,