How the work of Gian Carlo Rota had influenced my group research and life

How the w ork of Gian Carl o Rot a had influenced my group resear ch and life Andrzej Krzysztof Kwa ´ sniewski Member of the Institute of Combinatorics and its Applications High Sc hool of Mathematics and Applied Informatics Kamienna 17, PL-15-021 Bia lystok, Poland e-mail: kwandr@gmail.com Abstract: One outlines here in a b rief ove rview ho w th e work of Gian Carlo R ota h ad influenced my researc h and life, starting from the end of the last century up to present time state of The Internet Gian Carlo Rota Polish Seminar. Key W ords: ex tended um bral calculus, Gra ves-Heisen ber g-W eyl algebra, po sets, graded digra phs, AMS Class ification Numbers: 05A40, 81 S99, 06A06 ,0 5B20, 05 C7 affiliated to The Internet Gian-Car lo Polish Seminar: http://ii. uwb.e du.pl/akk/sem/sem r ota. htm 1 Ho w did I h ad come o v e r How and when I did ca me across the work of Gian Carlo Rota - this I do not remember. May b e it was only in 1997 becaus e of streams of thousands of r eferences o n the so ca lled q -deformations (extensions) that I was slightly inv olv ed in. Then I started to ag nize, to b e more fully aw are of the Gian Carlo Rota’s and his friends’ and disciples’ outstanding impo rtance with his and theirs mathematica l culture main stream inherited ideas, language and goals es pe cially ther e, where b oth analysis a nd c ombinatorics meet to enjoy the join into the alloy ore - the crystalline forma tion of Mathemagics. May it be then in December or so in 19 98 at Bia lystok - when I was m uch impressed by a s eries o f Professo r Oleg Vikto rovic h Visko v from Steklov Institute lectures on um bral calculus and all that. Since that time in almost a ll my ”‘umbra”’ articles I frequen tly r efer to P rofessor Visk ov co ntributions [1-4] and others - for more s ee [5-7]. Thes e [5-7] r eferences are examples of m y first contributions (including ”‘upside down notatio n”’) to the extended umbral ca lculus. What is this ”‘upside down notatio n”’ from [5 -7] ? It is just this : k F ≡ F k , where F is a natur al num bers v alued s equence. This no tation inspire d by Gauss and in the spirit of Knuth via the rea soning just repea ted with ” k F ” num bers replacing k - na tural num ber s leads one to transpar ent clean results in a lot of ca ses as for example in the rece n t acyclic dig raph’s ar ticles [8-10 ]. F or this notation see also Appendix in [1 1]. 2 Gra v es-Heisen b erg-W eyl algebra The ingenious ideas of differential a nd dual graded p osets t hat w e ow e to Stanley and F omin (see [10]) bring to gether combinatorics, representation theory , top ol- ogy , geometry a nd many more spec ific branches of mathematics and mathemat- 1 ical ph ysics thanks to intrinsic ingredient of these mathematical des criptions which is the Graves - Heisenber g - W eyl (GHW) algebra usually attributed to Heisenberg b y physicists a nd to Herman W eyl b y mathematicians and some- times to both of them. As notice d by Oleg Viktorovich Viskov in [4] the formula [ f ( a ) , b ] = cf ′ ( a ) where [ a, b ] = c, [ a, c ] = [ b , c ] = 0 per tains to Charles Grav es from Dublin [12]. Then it was re- discov ered b y Paul Adrien Ma urice Dirac and others in the next century . Let us then note that the picture that emer ges in [5-7] disclose s the fact that any um bral r epresentation of finite (extended) op era tor calculus or equiv alent ly - any umbral r epresentation of GHW algebr a mak es up a n example of the alge- braization of the analysis with gener alized differen tial ope rators of Marko wsky acting on the algebra of polynomia ls or other algebra s as for e xample formal series a lgebras . 3 Cob w eb p osets and D A Gs named KoD A Gs KoDA Gs are Hasse diag rams -hence dir ected a cyclic g raphs o f c obw eb partially ordered sets whic h are s ecluded in a natural w a y from mult i-ary relations ch ains’ digraphs. The family of these so ca lled cobw eb pos ets has b een inv en ted b y the author at the dawn of this century (for ear lier references see [13,1 4]- for the recent ones see [8-11 ]) . These structures are such a genera lization of the Fibo nacci tree growth that allo ws join t com binatorial inter pretation [13,1 4] for all of them under the co mbinatorial a dmissibility condition. Let { F n } n ≥ 0 be a natural num bers v alued sequence with F 0 = 1 (or F 0 ! ≡ 0! being exce ptional as in case of Fibonacci n um ber s). An y suc h sequence uniquely designates b oth F -nomial coefficients of an F -extended um bral calculus as well as F -cob w eb p o set defined in [1 3]. If these F - nomial co efficients are natural nu mbers or zero then we call the sequence F - the F - cob w eb admissi ble sequence . Definition 1 L et n ∈ N ∪ { 0 } ∪ {∞} . L et r , s ∈ N ∪ { 0 } . L et Π n b e t he gr a de d p artial or der e d set (p oset) i.e. Π n = (Φ n , ≤ ) = ( S n k =0 Φ k , ≤ ) and h Φ k i n k =0 c o nstitutes or der e d p artition of Π n . A gr ade d p oset Π n with finite set of minimal elements is c al le d c obweb p oset iff ∀ x, y ∈ Φ i.e. x ∈ Φ r and y ∈ Φ s r 6 = s ⇒ x ≤ y or y ≤ x, Π ∞ ≡ Π . See Fig .1. 2 Definition 2 L et any F - c o bweb admissible se quenc e b e given then F -nomial c o efficients ar e define d as fol lows  n k  F = n F ! k F !( n − k ) F ! = n F · ( n − 1) F · ... · ( n − k + 1) F 1 F · 2 F · ... · k F = n k F k F ! while n, k ∈ N and 0 F ! = n 0 F = 1 with n k F ≡ n F ! k F ! staying for fal ling factorial. Definition 3 C max (Π n ) ≡ { c = < x k , x k +1 , ..., x n >, x s ∈ Φ s , s = k , ..., n } i.e. C max (Π n ) is the set of al l maximal chains of Π n Definition 4 L et C max h Φ k → Φ n i ≡ { c = < x k , x k +1 , ..., x n >, x s ∈ Φ s , s = k , ..., n } . Then the C h Φ k → Φ n i set of H asse sub-diagr am c orr esp onding maximal chai ns defines biunivo quely the layer h Φ k → Φ n i = S n s = k Φ s as t he set of maximal chains’ no de s and vic e versa - for these gr ade d DA Gs (KoD AG s include d). The eq uiv ale n t to that of [13,1 4] formulation of combinatorial interpretation of cobw eb po sets via their cover relation digra phs (Hasse diagrams) is the following. Theorem (Kwa ´ sniewski) F or F -c o bweb admissible se quenc es F -nomial c o efficient  n k  F is the c ar dinality of the family of equip otent to C max ( P m ) mutual ly disjoi nt maximal chains sets, al l to gether p ar titioning the set of maximal chains C max h Φ k +1 → Φ n i of the layer h Φ k +1 → Φ n i , wher e m = n − k . F or environment needed and then simple com binatorial pr o of see [14,13] e asily accessible via Arxiv. One uses for that to pro of the graded structure of Hasse diag gram and the notion of the layer. Comment 1 . F or the ab ov e Kwa ´ sniewski co mbinatorial int erpretation o f F - nomials’ array the diag ram b eing directed or not do es no t matter o f cour se, as this combin atorial interpretation is equally v alid for partitions of the family of S impl eP ath max (Φ k − Φ n ) in co mparability graph of the Ha sse digraph with self-explanator y notation use d on the wa y . And to this end rec all: a po set is graded if a nd only if every co nnected co mp onent of its comparability graph is graded. W e are concerned here with connected gr aded graphs and digraphs. If one imp o ses further requirements with respec t F - sequences denominating bo th F -extended Um bral (Finite Ope rator) Calculus and cov er r elation dia- grams (Has se) of the corresp onding cob w eb p oset then further sp ecific prob- lems, their so lutions and sp ecific digraph-co mbin atorial in terpretations are ar - rived at. F or fresh results of the Studen t participant o f The In ternet Gian Carlo Rota P olish Seminar see [17]. F or his recent discov eries see [16,1 7]. 3 Figure 1: Display of th e la y er h Φ 1 → Φ 4 i = the subposet P 4 of th e F = G aussian int egers sequence ( q = 2) F -co bw eb p oset a nd σP 4 subpo set of the σ p ermuted Gaussian ( q = 2 ) F -cobw eb pose t . 4 Ho w all that had influenced m y and m y re- searc h group life ? The Gian Carlo Rota P olish Seminar has been tra nsformed in 2008 and is active now as The In ternet Gian Carlo Rota Polish Seminar: http://ii. uwb.e du.pl/akk/sem/sem r ota. htm . W e are contin uing the research. References [1] O.V. Visko v O p er ator char acterizatio n of gener alize d App el p oly nomials (in Russian) Dokl. Ak a d. Nauk SSSR 225 No 4 (1975) 749-752 , English trans - lation in Soviet Ma th. Dokl. 16 (1975 ) 152 1-152 4. [2] O.V. Visk ov On the b asis in the sp ac e of p o lynomials Dokl. Ak ad. Nauk SSSR 239 No 1 (19 78) 22 -25; Soviet Math. Dokl. 1 9 (1 978) 250 -253 . [3] O.V. Visko v Inversion of p ower series and L agr ange formula Dokl. Ak ad. Nauk SSSR 254 No 4 (1980) 769- 271 : Soviet Math. Dokl. 22 (198 0) 330-3 32. [4] O.V. Visko v ”On One Result of George Boole” (in Russian) In tegral T rans- forms and Special F unctions-Bulletin v ol. 1 No2 (1997) p. 2-7 [5] A.K.Kwa ´ sniewski T owar ds ψ -ext ension of Finite Op er ator Calculus of R ota Rep. M ath. Ph ys. 48 (3), 305-342 (2001). arXiv:math/04020 78 v1, [v1] Thu, 5 F eb 2 004 13 :02:30 GMT [6] A.K. Kwa ` sniewski O n exten de d finite op e r ator c alculus of R ota and quan- tum gr oups Integral T ransforms and SpecialF unctions 2 (4), 333 (2001) [7] A.K.Kwa ` sniewski Main the or ems of extende d finite op er ator c alculus Inte- gral T ransfor ms and Sp ecial F unctions 14 , 333 (2003 ). [8] A.K. Kwa ´ sniewski , Cobweb Posets and KoD A G Digr aph s ar e R epr esent ing Natur al Join of R elations, their di-Bigr ap hs and the Corr esp onding A dja- c ency Matric es , arXiv:ma th/0812 .4066v 1,[v1] Sun, 21 Dec 2 008 23 :04:48 GMT 4 [9] A.K. Kwa ´ sniewski , Some Cobweb Posets Digr aphs’ Elementary Pr op erties and Questions ar Xiv:0812 .4319 v1, [v1] T ue, 23 Dec 2008 0 0:40:41 GMT [10] A.K. Kwa ´ sniewski , Gr ade d p osets zeta matrix formula arXiv:0 901.0 155 v1 , [v1] Th u, 1 J an 200 9 01:43 :35 GMT [11] A. K. Kwa ´ sniewski, M. Dziemia ´ nczuk, On c obwe b p o sets’ most re levant c o dings , ar Xiv:0804.1 728 v1 [v1 ] Thu, 10 Apr 2008 15:09:2 6 GMT [12] Char les Grav es, On the princip les which r e gulate the inter change of symb ol s in c ertain symb o lic e quations Pro c. Roy al Irish Academ y vol. 6 , 1853-1 857 , pp. 1 44-1 52 [13] A. Krzysztof Kwa ´ sniewski, Cobweb p osets as nonc ommutative pr efabs , Adv. Stud. Con temp. 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