First Observations on Prefab Posets Whitney Numbers

First observ ations on Prefab p osets‘ Whitney n um b ers A. Krzysztof Kwa ´ sniewski the Dissident - relegated by Bia lystok Universit y authorities from the Institute of Computer Science to F acult y of Physics ul. Lipow a 41, 15 424 Bia lystok, Poland e-mail: kw andr@gmail.com Summary W e in tro duce a natural partial order ≤ in structurally natural finite subsets of the cobw eb prefabs sets recently constructed by the present author. Whitney num- b ers of t he second kind of the corresp onding subposet which constitute Stirling-like n umbers‘ triangular array - are then calculated and the explicit formula for them is pro vided. Next - in the second construction - we endow the set sums of prefabiants with suc h an another partial order that their their Bell-lik e num b ers include Fib onacci triad sequences introduced recently b y the present author in order to extend famous relation betw een binomial Newton coefficients and Fibonacci num b ers on to the infin- it y of their relatives among which there are also the Fibonacci triad sequences and binomial-lik e co efficients (incidence co efficients included). The first partial order is F - sequence indep endent while the second partial order is F -sequence dep enden t where F is the so called admissible sequence determining cobw eb poset by construction. An F -determined cobw eb p oset‘s Hasse diagram b ecomes Fib onacci tree sheathed with sp ecific cobw eb if the sequence F is c hosen to b e just the Fib onacci sequence. AMS Classification Numbers: 05C20, 11C08, 17B56 . Key W ords: prefab, exp o- nen tial structure, cobw eb p oset, Whitney num b ers, Bell-like n umbers, Fib onacci-like sequences presen ted (No vem b er 2006) at the Gian-Carlo Rota P olish Seminar http://ii.uwb.e du.pl/akk/sem/sem rota.htm published: Adv ances in Applied Clifford Algebras V olume 18 , Number 1 / F ebruary , 2008 , 57-73 ONLINE F IRST, Springer Link Date, F riday , August 10, 2007 1 In tro duction The clue algebraic concept of combinatorics - the so called prefab (with asso ciative and commutativ e comp osition) was in tro duced in [1], see also [2,3]. The elements of prefabs are called since now on - prefabiants. In [4] the present author had con- structed a new broader class of prefab‘s extending combinatorial structure based on the so called cob web p osets (see Section 1. [4] for the definition of a cobw eb poset as w ell as a combinatorial interpretation of its c haracteristic binomial-t yp e co efficien ts - for example- fib onomial ones [5,6]). Here we in tro duce tw o natural partial orders: one ≤ in grading-natural subsets of cob- w eb‘s prefabs sets [4] and in the second prop osal we endow the set sums of prefabiants with such another partial order that one may extend the Bell num b ers to sequences of Bell-lik e num b ers encompasing among infinity of others the Fib onacci triad sequences in tro duced by the presen t author in [7]. 2 Prefab based p osets and their Whitney n um- b ers. Let the family S of combinatorial ob jects ( pr ef abiants ) consists of all lay ers h Φ k → Φ n i , k < n, k , n ∈ N = 0 , 1 , 2 , ... and an empt y prefabian t i . The set ℘ of prime ob jects consists of all sub-posets h Φ 0 → Φ m i i.e. all P m ‘s m ∈ N constitute from no w on a family of prime pr ef abiants whic h w e define after[4]in t w o steps. Namely accompanying the set E of edges to the set V of vertices - one obtains the Hasse diagram where here down p, q , s ∈ N . (Conv ention: Edges stay for arro ws directed - say - up wards - see examples below). Definition 1 P = h V , E i , V = [ 0 ≤ p Φ p , E = {hh j, p i , h q , ( p +1) ii} [ {hh 1 , 0 i , h 1 , 1 ii} , w her e 1 ≤ j ≤ p F , 1 ≤ q ≤ ( p + 1) F . . The finite cob w eb sub-poset P m is then defined accodingly . Definition 2 P m = h V m , E m i , wher e V m = S 0 ≤ s ≤ m Φ s and E m is define d as E r estricte d to V m by 1 ≤ p ≤ m − 1 is c al le d the prime c obweb poset. La yer h Φ k → Φ n i is considered here to b e the set of all max-disjoint isomorphic copies (iso-copies) of P m , m = n − k [4]. As a matter of illustration we quote after [4] examples of cobw eb p osets‘ Hasse Diagrams [9] so that the lay ers become visualized. 2 Fig.1. Displa y of Natural num b ers‘ cobw eb poset. Fig.2. Displa y of Ev en Natural num b ers ∪{ 1 } -cob web p oset. 3 Fig3. Displa y of Odd natural num b ers‘ cob web p oset. Fig.4. Displa y of divisible by 3 natural n umbers ∪{ 1 } - cob w eb poset. 4 Fig.5. Displa y of Fib onacci num bers‘ cob web p oset. 3 Cob w eb p osets‘ combinatorial in terpretation As seen ab o ve - for example the F ig . 5. displa ys the rule of the construction of the Fi- b onacci ”cobw eb” p oset. It is b eing visualized clearly while defining this [ non-lattic e! ] cob web p oset P with help of its incidence matrix [8]. The incidence ζ function matrix represen ting uniquely just this cobw eb poset P has the staircase structure corresp on- den t with ” cobwebed ” Fib onacci T ree i.e. a Hasse diagram [9] of the particular partial order relation under consideration. 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 · · · 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 · · · 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 · · · 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 · · · 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 · · · 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 · · · 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 · · · 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 · · · 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 · · · 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 · · · 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 · · · 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 · · · 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 · · · . . . . . . . . . . . . . . . . . · · · 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Figure 6. The incidence matrix ζ for the Fib onacci cobw eb p oset Note The knowledge of ζ matrix explicit form enables one to construct (count) via standard algorithms [8] the M¨ obius matrix µ = ζ − 1 and other t ypical elements of incidence algebra perfectly suitable for calculating num b er of c hains, of maximal c hains etc. in finite sub-p osets of P . All elements of the corresponding incidence algebra are then giv en by a matrix of the Fig.6 with 1‘s replaced by an y reals ( or ring elemen ts in more general cases). W e ha ve a natural combinatorial ob ject characterizig the cob w eb posets Hasse directed graphs. Namely - in general ([4-7], [10], [12], [13]) - given an y sequence { F n } n ≥ 0 of nonzero reals one may define its corresp onding binomial-lik e F − nomial co efficients in the spirit of W ard‘s Calculus of sequences [13]s as follo ws Definition 3 „ n k « F = F n ! F k ! F n − k ! ≡ n k F k F ! , n F ≡ F n 6 = 0 , n ≥ 0 wher e we make an analo gy driven identific ations in the spirit of War d‘s Calculus of se quenc es (0 F ≡ 0) : n F ! ≡ n F ( n − 1) F ( n − 2) F ( n − 3) F . . . 2 F 1 F ; 0 F ! = 1; n k F = n F ( n − 1) F . . . ( n − k + 1) F . This is just the adaptation of the notation for the purpose Fib onomial Calculus case (see references in [4-7], [10], 12]). The crucial and elementary observ ation now is that the cobw eb p oset combinatorial in terpretation of F -binomial co efficients [4-7,10,12,14,16] makes sense not for arbi- trary F sequences as F − nomial co efficien ts should b e nonnegative integers. Definition 4 A se quenc e F = { n F } n ≥ 0 is c al le d c obweb-admissible iff „ n k « F ∈ N f or k , n ∈ N . 6 Righ t from the definition of P via its Hasse diagram here now follow quite obvious and imp ortan t observ ations. They lead us to a combinatorial interpretation of cob web p oset‘s c haracteristic binomial-like coefficients (for example - fib onomial ones [6,16]). Here they are with the first obvious observ ation at the start. 14, Observ ation 1 The numb er of maximal chains starting fr om The R o ot (level 0 F ) to r e ach any p oint at the n − th level with n F vertic es is e qual to n F !. Observ ation 2 ( k > 0) The numb er of al l maximal chains in-b etwe en ( k + 1) − th level Φ k +1 and the n − th level Φ n with n F vertic es is e qual to n m F , wher e m + k = n. Indeed. Denote the num b er of wa ys to get along maximal c hains from any fixed p oin t (the leftist for example) in Φ k to ⇒ Φ n , n > k with the symbol [Φ k → Φ n ] then ob viously w e hav e : [Φ 0 → Φ n ] = n F ! and [Φ 0 → Φ k ] × [Φ k → Φ n ] = [Φ 0 → Φ n ] . In order to formulate the combinatorial in terpretation of F − sequence − nomial coef- ficien ts ( F-nomial - in short) let us consider all finite ”max-disjoint” sub-p osets ro oted at the k − th lev el at an y fixed v ertex h r, k i , 1 ≤ r ≤ k F and ending at corresponding n umber of vertices at the n − th level ( n = k + m ) where the ”max-disjoint” sub-p osets are defined b elo w. Definition 5 Two families of maximal chains including two e quip otent c opies of P m ar e said to b e max-disjoint if c onsider e d as sets of maximal chains they ar e disjoint i.e they have no maximal chain in c ommon. (Al l P m ‘s c onstitute fr om now on a family of the so c al le d prime [4,10] pr ef abiants ). An e quip otent c opy of P m [‘ e quip-c opy ’] is define d as such a family of maximal chains e quinumer ous with P m set of maximal chains that the it c onstitues a sub-p oset with one minimal element. Definition 6 We denote the numb er of al l max-disjoint e quip otent c opies of P m r o ote d at any vertex h j, k i , 1 ≤ j ≤ k F of k − th level with the symb ol „ n k « F . One uses the customary conv ention: „ 0 0 « F = 1 . Naturally- let us recall- the ab ov e definition makes sense not for arbitrary F se- quences as F − nomial co efficients should b e nonnegative integers i.e. the sequence F = { n F } n ≥ 0 m ust be cob web-admissible. Problem 0. The partition or tiling problem. Supp ose no w that F is a cobw eb admissible sequence. Let us introduce σ P m = C m [ F ; σ < F 1 , F 2 , ..., F m > ] . 7 the equip oten t sub-p oset obtained from P m with help of a p ermutation σ of the se- quence < F 1 , F 2 , ..., F m > . Then P m = C m [ F ; < F 1 , F 2 , ..., F m > ] . Consider the lay er h Φ k → Φ n i , k < n, k , n ∈ N . La yer is considered here to b e the set of all max-disjoint equip otent copies of P n − k . The question then arises , whether and under which conditions the lay er may be partitioned with help of max- disjoin t blo cks of the form σ P m . At first - this main question answer is in affirmitav e. Some computer exp eriments done by student Maciej Dziemia ´ nczuk [17] are encourag- ing Ho wev er problems : ”how many?” or ”find it all” are still opened . Recall now that the n um b er of wa ys to reach an upper level from a low er one along an y of maximal chains i.e. the num b er of all maximal chains from the level Φ k +1 to ⇒ Φ n , n > k is equal to [Φ k → Φ n ] = n m F . Naturally then we ha ve „ n k « F × [Φ 0 → Φ m ] = [Φ k → Φ n ] = n m F (1) where [Φ 0 → Φ m ] = m F ! counts the n umber of maximal chains in any equip-cop y of P m . With this in mind w e see that the following holds. Observ ation 3 ( n,k ≥ 0 ) L et n = k + m . L et F b e any c obweb admissible se quenc e. Then the numb er of max-disjoint e quip-c opies i.e. sub-p osets e quip otent to P m , r oote d at the same fixe d vertex of k − th level and ending at the n-th level is e qual to n m F m F ! = „ n m « F = „ n k « F = n k F k F ! . Note The ab ov e Observ ation 3 pro vides us with the new combinatorial in terpre- tation of the class of all classical F − nomial co efficients including distinguished binomial or distinguished Gauss q - binomial ones or Konv alina generalized binomial co efficien ts of the first and of the second kind [11,12]- which include Stirling num- b ers to o. The v ast family of W ard-like [13] admissible by ψ = h 1 n F ! i n ≥ 0 -extensions F -sequences [12,14,16] includes also those desired here which shall be called ”GCD- morphic” sequences. This means that GC D [ F n , F m ] = F GC D [ n,m ] where GC D stays for Greatest Common Divisor op erator. The Fib onacci sequence is a muc h ontrivial [16,6] and guiding famous example of GCD-morphic sequence. Naturally incidence co efficien ts of any reduced incidence algebra of full binomial type [8] are GCD-morphic sequences therefore they are now indep endently given a new cob web com binatorial interpretation via Observ ation 3. More on that - see the next section where prefab com binatorial description is b eing serv ed. Before that - on the w ay - let us form ulate the following problem (opened?). Problem 1 Find effective char acterizations of the c obweb admissible se quence i.e. find al l examples. Note on admissibility . Observ ation 3 from [16] provides us with the new c ombina- torial interpretation of the class of all classical F − nomial coefficients including 8 distinguished binomial or distinguished Gauss q - binomial ones or Konv alina general- ized binomial coefficients of the first and of the second kind [11]- which include Stirling n umbers too. This v ast family of W ard-like [13] cob w eb admissible F -sequences - ad- missible at first by the so called ψ = h 1 n F ! i n ≥ 0 um bral extensions[9] - includes also those desired here whic h shall b e called ”GCD-morphic” sequences. Definition 7 The se quenc e of inte gers F = { n F } n ≥ 0 is c al le d the GCD-morphic se- quenc e if GC D [ F n , F m ] = F GC D [ n,m ] wher e GC D stays for Gre atest Common Divisor op er ator. Recall again : the Fib onacci sequence is a muc h nontrivial [6] and guiding example of GCD-morphic sequence . Naturally incidence co efficients of any reduced incidence algebra of full binomial type [8] are cobw eb-admissible. Question: which of these ab o v e are GCD-morphic sequences? In view of the Note on admissibilit y the following problems are apparently in ter- esting also on their own. Characterization Problem Find effe ctive characterizations of the c obweb admissi- ble se quenc e i.e. find al l examples. GCD-morphism Problem Find effe ctive char acterizations i.e. find al l examples. 4 Prefabs‘ Whitney n um b ers Consider then now the partially ordered family S of these lay ers considered to b e sets of all max-disjoint isomorphic copies (iso-copies) of prime prefabiants P m = P n − k as displa yed by Fig 1. - Fig.5. examples abov e. F or any F -sequence determining cobw eb p oset let us define in S the same partial order relation as follo ws. Definition 8 h Φ k → Φ n i ≤ h Φ k ♣ → Φ n ♣ i ≡ k ≤ k ♣ ∧ n ≤ n ♣ . F or con v enience reasons we shall also adopt and use the following notation: h Φ k → Φ n i = p k,n . The interv al [ p k,n , p k ♣ ,n ♣ ] is of course a subp oset of h S, ≤i . W e shall consider in what follo ws the subp oset h P k,n , ≤i where P k,n = [ p o,o , p k,n ] . 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 . Pro of: Obvious. Just dra w the picture {h l, m i , 0 ≤ l ≤ k ∧ 0 ≤ m ≤ n ∧ k ≤ n } of P k,n ‘ grid. Observ ation 2. The n um b er of maximal chains in h P k,n , ≤i is equal to the n um b er d ( k , n ) of 0 - dominated strings of binary i.e. 0 0 s and 1 0 s sequences d ( k , n ) = n + 1 − k n „ k + n n « . Pro of. The num b er we are lo oking for equals to the num b er of minimal walk-paths in [ k × n ] Manhattan grid [15] - paths restricted by the condition k ≤ n i.e. it equals to the n umber of 0 - dominated strings of 0 0 s and 1 0 s sequences. 9 Recall that ( d ( k, n )) infinite matrix‘s diagonal elements are equal to the Catalan n umbers C ( n ) C ( n ) = 1 n „ 2 n n « . as the Catalan num b ers count the num ber of 0 - dominated strings of 0 0 s and 1 0 s with equal num b er of 0 0 s and 1 0 s . Recall that a 0 - dominated string of length n is such a string that the first k digits of the string contain at least as many 0 0 s as 1 0 s for k = 1 , ..., n i.e. 0‘ s prev ail in appearance, dominate 1‘ s from the left to the righ t end of the string. 0 - dominated strings corresp ond bijectively to minimal b ottom - left corner to the right upper corner paths in an integer grid Z ≥ × Z ≥ rectangle part called Manhattan [15] with the restriction imp osed on those minimal paths to ob ey the ”safet y” condition k ≤ n . Commen t 1. Observ ation 2. equips the p oset h P k,n , ≤i with clear cut combinatorial meaning. The poset h P k,n , ≤i is naturally graded. h P k,n , ≤i p oset‘s maximal c hains are all of equal size (Dedekind property) therefore the rang function is defined. Observ ation 3. The rang r ( P k,n ) of P k,n = num b er of elements in maximal c hains P k,n min us 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 . Pro of: obvious. Just draw the picture {h l, m i , 0 ≤ l ≤ k ∧ 0 ≤ m ≤ n ∧ k ≤ n } of P k,n ‘ grid and note that maximal means paths without at a slan t edges. Accordingly Whitney n um b ers W k ( P l,m ) of the second kind are defined as follows (asso ciation: n ↔ h l , m i ) Definition 9 W k ( P l,m ) = X π ∈ P l,m ,r ( π )= k 1 ≡ S ( k , h l , m i ) . Here no w and afterwords we identify W k ( P l,m ) with S ( k , h l, m i ) called and view ed at as Stirling - lik e n um b ers of the second kind of the naturally graded p oset h P k,n , ≤i - note the association: n ↔ h l , m i . Righ t no w challenge problems. I. I. Let us define now Whitney num b ers w k ( P l,m ) of the first kind as follo ws (association: n ↔ h l, m i . Note the text-bo ok notation for M¨ obius function µ ) Definition 10 w k ( P l,m ) = X π ∈ P l,m ,r ( π )= k µ (0 , π ) ≡ s ( k , h l, m i ) . Here now and afterwards w e identify w k ( P l,m ) with s ( k , h l, m i ) called and viewed at as Stirling - like n umbers of the first kind of the p oset h P k,n , ≤i - note the asso ciation: n ↔ h l, m i . Problem 1 Find an explicit expression for w k ( P l,m ) ≡ s ( k , h l , m i ) =? and W k ( P l,m ) ≡ S ( k , h l, m i ) =? 10 Occasionally note that S ( k , h l , m i ) equals to the num ber of the grid p oints counted at a slant (from the up-left to the right-do wn) accordingly to the l + m = k requiremen t. Problem 2 Find the recurrence relations for w k ( P l,m ) ≡ s ( k , h l , m i and W k ( P l,m ) ≡ S ( k , h l, m i ) . W e define now (note the asso ciation: n ↔ h l , m i ) the corresp onding B ell-lik e n umbers B ( h l , m i ) of the naturally graded p oset h P k,n , ≤i as follows. Definition 11 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) . Pro of: Just draw the picture {h l , m i , 0 ≤ l ≤ k ∧ 0 ≤ m ≤ n ∧ k ≤ n } of P k,n ‘ grid and note that S ( k , h l , m i ) equals to the n um b er of the grid p oints coun ted at a slant (from the up-left to the right-do wn) accordingly to the l + m = k requiremen t. Summing them up o ver all giv es the size of P k,n . Commen t 2. Observ ation 4. equips the p oset‘s h P k,n , ≤i Bell-like num b ers B ( h l , m i ) with clear cut com binatorial meaning. 5 Set Sums of prefabian ts‘ p osets and their Whit- ney n umbers. In this part we consider prefabiants‘ set sums with an appropriate another partial order so as to arrive at Bell-like num bers including Fib onacci triad sequences introduced recen tly b y the present author in [16] - see also [7,6]. Let F b e an y ”GCD-morphic” sequence. This means that GC D [ F n , F m ] = F GC D [ n,m ] where GC D stays 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 giv en by the sum b elo w. Definition 12 P ( n, F ) = [ 0 ≤ p h Φ p → Φ n − p i = [ 0 ≤ l P n − l with the partial order relation defined for n − 2 l ≤ 0 according to Definition 13 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 introduce accordingly . Definition 14 W k ( P n,F ) = X π ∈ P n,F ,r ( π )= k ≡ S ( n, k , F ) . 11 Righ t from the definitions ab ov e w e infer that: (recall that rang of P l is l .) Observ ation 5. W k ( P n,F ) = X π ∈ P n,F ,r ( π )= k ≡ S ( k , n − k , F ) = „ n − k k « F . Here now and afterwords w e identify W k ( P n,F ) = S ( n, k , F ) viewed at and called as Stirling - like num bers of the second kind of the P defined in [10]. P by construction (see Figures abov e) displa ys self-similarity prop erty with respect to its prime prefabi- an ts sub- p osets P n = P ( n, F ). Righ t no w challenge problems. II. W e rep eat with obvious replacemen ts of corresp onding sym b ols, names and definitions the same problems as in ”Righ t no w challenge problems. I”. Here now consequen tly - for any GC D -morphic sequence F (see: [10]) we define the corresp onding Bell-like num bers B n ( F ) of the p oset P ( n, F ) as follo ws. Definition 15 B n ( F ) = X k ≥ 0 S ( n, k , F ) . Due to the inv estigation in [7,16] we hav e right now at our disp osal all corresp onding results of [16,7] as the following identification with sp ecial 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 Definition 2.2. from [7]. Compare also with the sp ecial case of formula (6) in [16]. Recurrence relations. Recurrence relations for h α, β , γ i - Fibonacci sequences F [ α,β = ,γ ] n are to b e found in [7] - formula (9). Compare also with the sp ecial case form ula (7) in [16]. Closing-Op ening Remark. The study of further prop erties of these Bell-like num- b ers as well as the study of consequences of these identifications for the domain of the widespread data t ypes [7] and p erhaps for ev en tual new dynamical data t yp es we lea ve for the p ossibly coming future. Examples of sp ecial cases - a bunch of them - one finds in [7] containing [16] as a sp ecial case. As seen from the identification Observ ation 6. the sp ecial cases of h α, β , γ i - Fib onacci sequences F [ α,β ,γ ] n gain additional with resp ect to [16,7] combinatorial interpretation in terms Bell-lik e n um b ers as sums ov er rang = k parts of the poset i.e. just sums of Whitney n umbers of the poset P ( n, F ). This adjectiv e ”additional” shines brigh tly o ver Newton binomial connection constants b et w een bases h ( x − 1) k i k ≥ 0 and h x n i n ≥ 0 as these are Whitney n umbers of the num- b ers from [ n ] c hain i.e. Whitney n umbers of the p oset h [ n ] , ≤i . F or other elementary ”shining brigh tly” examples see Joni , Rota and Sagan excellent presentation in [18]. 6 On applications of new cob w eb p osets‘ origi- nated Whitney n um b ers Applications of new cobw eb p osets‘ originated Whitney num b ers such as extended Stirling or Bell num b ers are exp ected to b e of at least such a significance in appli- cations to linear algebra of formal series [linear algebra of generating functions [19]] as Stirling and Bell num b ers or their q -extended corresp ondent already are in the so called coheren t ph ysics [20] ( see [20] also for abundan t references on the sub ject). Also 12 straigh tforward applications of prefabs to coherent physics [20] are on line. [Quan tum coheren t states ph ysics is of course a linear theory with its principle of states‘ sup er- p osition]. In order to say more on the sub ject of this section and give some examples let us remind the equiv alence of exp onential structures by Stanley [21] with corresp onding exp onen tial prefabs [1]. In this context the let us indicate the crucial ”W ard‘ian - prefab‘ian” example we ow e to Gessel [22] with his q -analog of the exponential formula as expressed b y the Theo- rem 5.2 from [22]. W e also recall that the q -analog of the Stirling num b ers of the second kind inv esti- gated by Morrison in Section 3 of [23] constitute the same example of W ard‘ian - prefab‘ian extension as in the Bender - Goldman - W agner W ard - prefab example. As noticed there b y Morrison the ( γ − e.g .f . ) prefab exp onential form ula may equally w ell b e derived from the corresponding Stanley‘s exp onen tial formula in [21]. Let us then now come o v er to these exp onential structures of Stanley with an exp ected im- pact on the current considerations ( for definitions, theorems etc. see [21]). In this connection we recall quoting (notation from [21]) an imp ortant class of Stanley‘s Stir- ling - like num b ers S nk M ( n ) of the second and those of the first kind Stanley‘s Stirling - like num bers s nk M ( n ) . Both kinds are characteristic immanent for counting of exp o- nen tial structures (or equiv alently - corresp onding exp onential prefabs) and inheriting from there their combinatorial meaning. This is due to the fact [21] that ”with each exp onen tial structure is asso ciated an ”exp onential form ula” and more generally a ”con volution formula” which is an analogue of the w ell kno wn exp onential formula of en umerative combinatorics” [21]. Consequently with each exp onen tial structure are asso ciated Stirling-like , Bell-like n umbers and Dobinski - like formulas are exp ected also, of crucial impotance for generalized coheren t states‘ physics. As for the another examples let us consider in more detail exp onential structures. Exp onen tial structures. Let { Q n } n ≥ 0 b e an y exponential structure and let { M ( n ) } n ≥ 0 b e its denominator sequence i.e. M ( n ) = n um b er of minimal elemen ts of Q n . Let | Q n | b e the n umber of elements of the p oset Q n | Q n | = X π ∈ Q n 1 . Example: F or Q = h Π n i n ≥ 1 where Π n is the partition lattice of [ n ] we ha ve M ( n ) = 1. Define ”Whitney-Stanley” num ber S n,k to b e the num ber of π ∈ Q n of degree equal to k ≥ 1 i.e. S n,k = X π ∈ Q n , | π | = k 1 . Define S n,k - generating characteristic p olynomials (vide exponential p olynomials) in standard wa y W n ( x ) = X π ∈ Q n x | π | = n X k =1 S n,k x k . Then the exp onen tial formula ( W 0 ( x ) = 1 = M (0)) b ecomes ∞ X n =0 W n ( x ) y n M ( n ) n ! = exp { xq − 1 ( y ) } , where q − 1 ( y ) = ∞ X n =1 y n M ( n ) n ! ≡ exp ψ − 1 , 13 with the obvious identification of ψ -extension choice here. Hence the p olynomial se- quence h p n ( x ) = W n ( x ) M ( n ) i n ≥ 0 constitutes the sequence of binomial p olynomials i.e. the basic sequence of the corresp onding delta operator ˆ Q = q ( D ). W e observ e then that p n ( x ) = n X k =0 S n,k x k M ( n ) ≡ n X k =0 [0 , 1 , 2 , ..., k ; b n ] x k are just exp onential p olynomials‘ sequence for the equidistan t no des case i.e. Newton- Stirling n umbers of the second kind S ∼ n,k ≡ S n,k M ( n ) . Both num b ers and the exp onential sequence are b eing bi-univocally determined by the exponential structure Q . This is a sp ecial case of the one considered in [20] and w e hav e the - what we call- Newton- Stirling-Dobinski f orm ula (notation, history and details- see [20]) p n ( x ) = 1 exp( x ) ∞ X k =0 b n ( k ) x k k ! = n X k =0 [0 , 1 , 2 , ..., k ; b n ] x k , ( N − S − D ob ) where h b n i n ≥ 0 is defined by b n ( x ) = n X k =0 S ∼ n,k x k . Note the iden tification b n ( x ) = w n ( x ) M ( n ) , where w n ( x ) = − X π ∈ Q n µ ( ˆ 0 , π ) λ | π | . µ is M¨ obius function and ˆ 0 is unique minimal element adjoined to Q n . Corresp onding Bell-like num bers [20] are then giv en b y p n (1) = 1 exp( x ) ∞ X k =0 b n ( k ) k ! = n X k =0 [0 , 1 , 2 , ..., k ; b n ] , ( N − S − B ell ) . Besides those ab o ve - in Stanley‘s paper [21] there are implicitly presen t also in verse- dual ”Whitney-Stanley” num b ers s n,k of the first kind i.e. s n,k = − X π ∈ Q n , | π | = k µ ( ˆ 0 , π ) . On this occasion and to the end of considerations on exponential structures and Stir- ling like num b ers let us make few remarks. q -extension of exp onen tial formula applied to en umeration of p ermutations b y inv ersions is to be find in Gessel‘s pap er [22] (see there Theorem 5.2.) where among others he naturally arrives at the q -Stirling num- b ers of the first kind giving to them combinatorial in terpretation. Recent extensions of the exp onen tial formula in the prefab language [1] are to b e find in [4]. Then note : exp onen tial structures, prefab exp onential structures (extended ones - included) i.e. sc hemas where exponential formula holds-imply the existence of Stirling like and Bell lik e num b ers. As for the Dobinski-lik e form ulas one needs binomial or extended bino- mial coefficients‘ conv olution as it is the case with ψ -extensions of umbral calculus in its operator form. Other Generalizations in brief. W e indicate here three kinds of extensions of Stirling and Bell n umbers - including those whic h appear in coheren t states‘ appli- cations in quan tum optics on one side or in the extended ro ok theory on the other side. In the supplemen t for this brief account to follo w on this topics let us note that apart from applications to extended coherent states‘ ph ysics of quantum oscillators or strings [6 - 11, 24, 25] and related F eymann diagrams‘ description [26] where we 14 face the sp ectacular and inevitable emergence of extended Stirling and Bell n umbers (consult also [27]) there exists a go od deal of work done on discretiz ation of space - time [28] and/or Sc hrodinger equation using umbral methods [29] and GHW algebra represen tations in particular (see: [28, 29] for references). Ac kno wledgemen ts Discussions with Participan ts of Gian-Carlo Rota Polish Seminar on all related topics http://ii.uwb.e du.pl/akk/sem/sem rota.htm - are appreciated with pleasure. 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