Comments on combinatorial interpretation of fibonomial coefficients - an email style letter

Commen ts on com binatorial in terpretation of Fib onomial co efficien ts- an e-mail st yle letter (*) A.K.Kw a ´ sniewsk i High Sc ho ol of Mathematics and Applied Informatics PL - 15-021 Bialy stok , ul.Kamienna 17, P oland e-mail: kw andr@u wb.edu.pl April 25, 2022 (*) Bulletin of the Institute of C om binatorics and its Applica- tions v ol. 42 ( 2004 ) : 10-11. Presen ted also at the Gian-Carlo Rota P olish Seminar http://ii.uwb.e du.pl/akk/sem/ sem r ota.htm I . Up to our kno wledge -since ab out 126 years we we re lac king of clas- sical typ e com binatorial in terpretation of Fib onomial co efficie nts as it wa s Luk as [ 1] - to our kno wledge -who w as the first wh o h ad defi n ed Finon o- mial co effici ents and deriv ed a recurrence for them (see Historical Note in [2] ). Namely as accurately noticed by Knuth and Wilf in [3] the r ecur - ren t relations for Fib onomial co efficien ts ap p eared already in 1878 Lu k as w ork [1] and in our opinion - Lu ca s Thorie des F onctio ns Numriques S im - plemen t Prio diques is the far more non-accident al con text for bin omia l and binomial-t yp e co efficie nt - Fib onomial co effici ents includ ed. I I . Recen tly [4 ] Kw a ´ sniewski com binatorial interpretatio n of Fib onomial co effici ents has b een prop osed in th e spirit [2] of the historica lly classical standard in terpretations according to the sc hematic corresp ondences: SETS —SUBSETS (without and with rep etitions)— Binomial coefficient —i.e. we are d eal ing with LA TTICE of su bsets SET P AR TITIONS: Stirling num b ers of the second kind — num b er of partitions into exactly k blo cs - i.e. w e are dealing with LA TTICE of par- titions. PERMUT A TION P AR TI T IONS : Stirling n umber s of th e fir st kind — 1 n umb er of p ermutat ions con taining exactly k CYCLE S . SP A CE S : q -Gaussian co efficien t — num b er of k -d imensional subspaces i n n-th dimensional space o ve r Galois field GF ( q ) — i.e. w e are dealing with LA T TICE of s ubspaces. (F or non trivial and fruitful Konv alina‘s u nified in terpretation of the Binomial Co efficien ts, the S tirling Numb ers, and the Gaussian Co efficien ts see [5] ). POSET — Fib onomial co efficien ts — n umb er of corresp onding (see: [4, 2] ) fin ite ”cob web” subp osets of the so called ”cob web” p oset. I I I . A t the time of p ublishing [4] K w a ´ sniewski w as n ot aw are of the existence of the relev an t prepr in t [6] of Ira M. Gessel and X. G. Viennot (Just few hours ago I h a ve noticed this article via Go ogle ) There righ t after the Th eorem 25 (see Section 10 , p age 24 in [6] ) relating the num b er N(R) of nonin tersecting k -paths to Fib onomial coefficient s (via q -w eigh ted t yp e coun ting formula) the authors express their wish - w orthy to b e quoted: ” it would b e nic e to have a mor e natur al interpr etation then the one we have given ”..... ” R. Stanley has aske d if ther e is a binomial p oset asso ciate d with the Fib onomial c o e ffic i ents...” - W ell. The cobw eb lo cally finite infi n ite p oset b y Kwa ´ sniewski from [2, 4] is n ot of b inomial type. Ev en more ; th e inciden ce algebra origin argumen ts seem to make us not to exp ec t bin omia l type p oset come int o the game [7]. Am I righ t? An immediate question arises - what is the relation lik e b et wee n t hese t w o: Gessel and Viennot [6] and [4] p oints of v iew? W e shall try to elab orate more on that so on. References [1] Ed uard Lucas Thorie des F onction s Numrique s Simplement Prio diques American Journ al of Mathematics 1 (1878) : 184-240 (T ranslated from the F renc h by Sid ney Kra vitz , Edited by Douglas Lind Fib onacci As- so ciat ion 1969) [2] A.K.Kwasniewski Combinatoria l derivation of the r e curr enc e r elation for fib onomial c o efficients ArXiv: math.CO/040301 7 v1 1 Marc h 2004 [3] D. E. Knuth, H. S. Wilf The Power of a Prime that Divides a Gener al- ize d Binomial Co efficient J. Reine Angev. Math. 396 (1989) : 212-21 9 [4] A. K. Kwasniewski Information on c ombinatorial interpr etation of Fi- b onomial c o efficients Bull. S oc. S ci. Lett. Lo dz Ser . Rec h. Deform. 53, 2 Ser. Rec h.Deform. 42 (2003): 39-41 , ArXiv: math.CO/0402291 v1 18 F eb 2004 [5] John Kon v alina A Unifie d Interpr etation of the Binomial Co efficients, the Stirling Numb er, and the Gaussian Co efficients ,Amer. Math. Mon thly 245 107 (2000) : 901-910 [6] Ir a M. Gessel, X. G. Viennot Determinant Paths and Plane Partitions preprint (1992) http://cit eseer.nj.nec.com/gessel89 determ inan ts.html [7] A.K.Kwasniewski The se c ond p art of on duality triads‘ p ap er-On fib ono- mial and other triangles versus duality triads Bull. So c. Sci. Lett. Lo dz Ser. Rec h. Deform. 53, Ser. Rec h. Deform. 42 (2 003): 27 -37 ArXiv: math.GM/040 2288 v1 18 F eb. 2004 3