New type Stirling like numbers - an email style letter

New t yp e Stirling lik e n um b ers - an email st yle letter A. Krzysztof Kwa ´ sniewski High Sc ho ol of Mathematics and Applied Informatics Kamienna 17, PL-15-021 Bia lystok, Poland e-mail: kwandr@wp.pl AMS Classification Numbers: 05C20, 11C08, 17B56 . Key W ords: prefab, cob web poset, Whitney num bers, Fib onacci like sequences presen tation (No vem ber 2005 ) at the Gian-Carl o Rota Polish Seminar http://ii. uwb.e du.pl/a kk/sem/sem r ota.htm The not ion of the Fibonacci cob web poset from [1] has b een na turally extended to any ad- missible sequence F in [2] where it was also recognized that the celebrated pr efab notion of Bender and Goldman [3] - (see also [4,5]) - admits suc h an extension so as to encompass the new t yp e combinat orial ob jects from [2] as leading examples. Recently the presen t author had int ro duced also [6] t wo natural partial orders in there: one ≤ in grading-natural subsets of cob web ‘s prefabs sets [2] and in the second prop osal one endo ws the set sums of the so called ”prefabian ts” with such anot her partial order that one arrives at Bell-like n um b ers including Fib onacci triad sequences int ro duced by the present author in [ 7]. Here we quote the basic observ ations concerning the new type Stirling l ik e numbers as they appear in [ 6]. F or more on notation, Stirli ng lik e num bers of the first kind and for pro ofs - see [6]. The o v erall F -indep endent cl ass of p.o. set structure. Let the family S of com binatorial ob jects ( pr ef abiants ) consists of all lay ers h Φ k → Φ n i , k < n, k , n ∈ N ∪ { 0 } ≡ Z ≥ and an empt y prefabiant i . The set ℘ of pr ime ob jects consists of all sub- posets h Φ 0 → Φ m i i.e. all P m ‘s m ∈ N ∪ { 0 } ≡ Z ≥ constitute from no w on a family of prime pref ab iants [ 2]. La yer is considered here to b e the set of all max-disj oin t is omorphic copies (iso-copies) of P m = P n − k [2]. Consider then now the partially ordered family S of these lay er s considered to b e sets of all max-disj oin t i somorphic copies (is o-copies) of pr ime prefabiant s P m = P n − k . F or any F -sequence determining cobw eb p oset [ 2] let us define in S the same partial order relation as follows. Definition 1 h Φ k → Φ n i ≤ h Φ k ♣ → Φ n ♣ i ≡ k ≤ k ♣ ∧ n ≤ n ♣ . F or conv enience reasons we shall also adopt and use the following notat ion: h Φ k → Φ n i = p k,n . In what follows w e shall consider the subp oset h P k,n , ≤ i where P k,n = [ p o,o , p k,n ] . Then according to [6] we observ e the following. Observ ation 1. The size | P k,n | of P k,n = |{h l , m i , 0 ≤ l ≤ k ∧ 0 ≤ m ≤ n ∧ k ≤ n }| = ( n − k )( k + 1) + k ( k +1) 2 . Observ ation 2. The num b er of maximal chains i n h P k,n , ≤ i is equal to the num b er d ( k , n ) of 0 - domi nated strings of binary sequences d ( k , n ) = n + 1 − k n  k + n n  . Recall that ( d ( k , n )) infinite matrix‘s diagonal el emen ts are equal to the Catalan n umbers C ( n ) . The p oset h P k,n , ≤ i i s naturally graded. h P k,n , ≤ i p oset‘s maximal cha ins ar e all of equal size (Dedekind -Jordan prop ert y) therefore the rang function is defined. Observ ation 3. The rang r ( P k,n ) of P k,n = n umber of elemen ts i n maximal chains P k,n minus one = k + n − 1 . The rang r ( p l,m ) of π = p l,m ∈ P k,n is defined accordingly: r ( p l,m ) = l + m − 1 . Accordingly Whitney num bers W k ( P l,m ) of the second kind ar e defined as follows (asso cia- tion: n ↔ h l , m i ) Definition 2 W k ( P l,m ) = X π ∈ P l,m ,r ( π ) = k 1 ≡ S ( k , h l , m i ) . W e shall identify W k ( P l,m ) with S ( k , h l , m i ) called and viewed at as Stirling - l ik e n um b ers of the second kind of the naturally graded poset h P k,n , ≤ i . Let us define also the corresponding Bell-l ik e num b ers B ( h l, m i ) of the naturally graded p oset h P k,n , ≤ i . Definition 3 B ( h l , m i ) = l + m X k =0 S ( k , h l , m i ) . Observ ation 4. B ( h l , m i ) = | P l,m | = k ( k + 1) 2 + ( n − k )( k + 1) . The F -dep enden t, F -lab elled class of p.o . set structures. Let us consider no w prefabian ts‘ set sums with an appropriate another partial order so as to arr iv e at Bel l- like n umbers including Fib onacci triad sequences int ro duced recen tly by the presen t author in [7]. Let F b e any ”GCD-morphic” sequence [2]. This means that GC D [ F n , F m ] = F GC D [ n,m ] where GC D sta ys for Greatest Common Divisor mapping. W e define the F - dep endent finite partial ordered set P ( n, F ) as the set of prime prefabiants P l give n by the sum b elo w. Definition 4 P ( n, F ) = [ 0 ≤ p h Φ p → Φ n − p i = [ 0 ≤ l P n − l with the partial order r elation defined for n − 2 l ≤ 0 according to Definition 5 P l ≤ P ˆ l ≡ l ≤ ˆ l, P ˆ l , P l ∈ h Φ l → Φ n − l i . Recall that rang of P l is l . Note that h Φ l → Φ n − l i = ∅ for n − 2 l ≤ 0 . The Whitney n umbers of the second kind are i n tro duce accordingly . Definition 6 W k ( P n,F ) = X π ∈ P ( n,F ) ,r ( π )= k 1 ≡ S ( n, n − k , F ) . Right f rom the definitions ab o ve we infer that: Observ ation 5. W k ( P n,F ) = X π ∈ P ( n,F ) ,r ( π )= k 1 ≡ S ( n, n − k , F ) =  n − k k  F . 2 Referring to the cl assical examples from [ 8] we i den tify W k ( P ( n, F )) with S ( n, n − k , F ) called the Stirling - li k e num bers of the s econd kind of the P . P by construction displays self-simi larity pr operty with resp ect to its pr ime prefabian ts sub - p osets P n = P ( n, F ) . Consequen tly for an y GC D -morphic sequence F (see: [2]) we define the corresp onding Bell- like num b ers B n ( F ) of the p oset P ( n, F ) as follows. Definition 7 B n ( F ) = X k ≥ 0 S ( n, k , F ) . Due to the inv estigation in [7] we hav e right no w at our disp osal all corresp onding results of [7] as the following iden tification with s pecial case of h α, β , γ i - Fib onacci sequence h F [ α,β ,γ ] n i n ≥ 0 defined in [7] holds. Observ ation 6. B n ( F ) ≡ F [ α =0 ,β =0 ,γ =0] . n +1 Pro of: See the D efinition 2.2. from [7]. Recurrence relations. Recurrence r elations for h α, β , γ i - Fi bonacci sequences F [ α,β = ,γ ] n are to b e found in [7] - f ormu la (9). Compare also with the sp ecial case formula (7) in [9]. Remark. As seen fr om the identifica tion Observ ation 6. the special cases of h α, β , γ i - Fib onacci s equenc es F [ α,β ,γ ] n gain additional with resp ect to [7] combinato rial int erpre- tation in terms Bell-li k e num bers as j ust sums of Whitney n umbers of the p oset P ( n, F ) . This adjectiv e ”additional” applies sp ectacularly to N ewton binomial connection constan ts betw een bases h ( x − 1) k i k ≥ 0 and h x n i n ≥ 0 as these are Whitney num bers of the nu mbers from [ n ] chain i.e. Whitney num bers of the p oset h [ n ] , ≤i . F or other element ary ”shini ng bright ly” examples see Joni , R ota and Sagan excellen t pr esen tation in [8]. References [1] A. K. Kwa ´ sniewski, Comments on c ombinatoria l int erpr etation of fib onomial c o efficients - an email style letter Bulletin of the Institute of Combinat orics and its Applications 42 (2004), 10-11. [2] A. K. Kw a ´ sniewski, Cobweb p osets as nonc ommutative pr efabs submitted for pub- lication ArXiv : m ath.CO/0503 286 (2005) [3] E. Bender, J. Goldman Enum er ative u ses of gener ating functions , Indiana Univ. Math.J. 20 1971), 753-765. [4] D. F oata and M. Sch” utzen b erger, Th’eorie g’eometrique des p olynomes euleriens, (Lec- ture Notes in Math., No. 138). Spr inger-V erlag, Berlin and New Y ork, 1970. [5] A. Nijenhuis and H. S. Wilf, Combinato rial Al gorithms, 2nd ed., Academic Press, New Y ork, 1978. [6] A.K. Kwa ´ sniewski, Pr efab p osets‘ Whitney nu mb ers Ar Xiv: math.CO/051002 7 , 3 Oct 2005 [7] A. K . Kwa ´ sniewski, Fib onac ci-triad se quenc es Adv an. Stud. Con temp. Math. 9 (2) (2004),109 -118. [8] S.A. Joni ,G. C. Rota, B . Sagan F r om sets to funct ions: thr e e elementary exam- ples Discrete Mathematics 37 (1981), 193-2002. 3