On Characteristic Polynomials of the Family of Cobweb Posets

On Characteristi c P olynomia ls of the F amily of Cob w eb P osets Ew a Krot-Sieniaw sk a Institute of Computer Scienc e, Bia lystok Univ ersit y PL-15-887 Bia lystok, ul.Sosnow a 64, POLAND e-mail: ew akrot@wp.pl, ew akrot@ii.u wb.edu.pl Abstract This note is a resp onse to one of problems posed by A.K. Kw a ´ sniewski in [7]. Namely with { P n } n ≥ 0 being the sequence of finite cobw eb subpo sets, the looked fo r explic it formulas fo r corr esp onding s e quence { χ n ( t ) } n ≥ 0 of P n ’s characteristic polynomials are discov e r ed and deliv- ered here. The recurrence relation defining arbitrary family { χ n ( t ) } n ≥ 0 is also derived. KEY W ORDS: cobw eb pos et, the M¨ obius function of a poset, Whitney n umber s , characteristic polynomials . AMS Classification num b ers : 06A06, 06A07, 06A11, 11C08, 11B37 Presented at Gia n-Carlo Rota Polish Seminar: http://ii.uwb.edu.p l/akk/ s em/sem rota.htm 1 Cob w eb p osets The family of the so calle d co bw eb po sets Π has b een inv ent ed by A.K.Kwa ´ sniewski few years ago (for re fer ences se e : [5, 6]). These structures ar e such a gene r alization of the Fib onacci tree growth that allows join t combinatorial interpretation for all of them under the admissibility c ondition (see [7, 8]). Let { F n } n ≥ 0 be a natural n um ber s v alued sequence with F 0 = 1 (with F 0 = 0 being exceptional as in case of Fib ona cci num b ers). An y sequence sa tisfying this prop erty uniquely desig nates cobw eb p oset defined as follows. F or s ∈ N 0 = N ∪ { 0 } le t us to define le vels of Π: Φ s = {h j, s i , 1 ≤ j ≤ F s } , (in case of F 0 = 0 level Φ 0 corres p o nds to the empty ro ot {∅} ). ) Then 1 Definition 1. 1. Corr esp onding c obweb p oset is an infi nite p artial ly or der e d set Π = ( V , ≤ ) , wher e V = [ 0 ≤ s Φ s ar e the elements ( vertic es) of Π and the p artial or der r elation ≤ on V for x = h s, t i , y = h u, v i b eing elements of c obweb p oset Π is define d by formula ( x ≤ P y ) ⇐ ⇒ [( t < v ) ∨ ( t = v ∧ s = u )] . Obviously any cob web pose t can b e repr e s ented, v ia its Hasse diag ram, as infinite directed gra f Π = ( V , E ), where set V o f its v ertices is defined as above and E = { ( h j, p i , h q , ( p + 1) i ) } ∪ { ( h 1 , 0 i , h 1 , 1 i ) } , where 1 ≤ j ≤ F p and 1 ≤ q ≤ F ( p +1) stays for set o f (directed) edges. F or example the Has se diagram of Fibona c ci cobw eb poset designated by the famous Fibo nacci sequence is presented b elow. andso-up andso-up F=0 0 F=1 1 F=1 2 F=2 3 F=3 4 F=5 5 F=8 6 Fig. 1. The construction of the Fibonacci co bw eb pos et 2 The Kwasniewski co bw eb p o sets under co nsideration represented by gr aphs a re ex- amples of o derable directed a cyclic gra phs (oDA G) which we start to ca ll from now in brief: KoD AGs. These are s tructures of universal impo rtance for the w ho le of mathematics - in particula r for discrete ”‘mathemagics ”’ [h ttp:// ii.uwb.edu.pl/akk/ ] and computer sciences in genera l (quo tation from [7, 8] ): F or a ny given natura l n um ber s v alued sequence the graded (layered) cobw eb p osets‘ DA Gs are equiv a lently representations of a chain of bi- nary re lations. Every relation of the cobw eb p oset ch ain is biunivoca lly represented by the uniquely desig nated complete bipar tite digraph-a digraph whic h is a di-biclique designated b y the very giv en sequence. The cobw eb p oset is then to b e identified with a chain of di-bicliques i.e. by definition - a chain of complete bipar tite one direction digra phs. An y chain of rela tions is therefor e obtainable from the cobw eb pos et chainof complete rela tio ns v ia deleting arcs (arr ows) in di-biclique s . Let us underline it again : any chai n of r elations is obtainable fr om the c obweb p oset chain of c omplete r elations via deleting ar cs in di- bicliques of t he c omplete r elations chain. F or tha t to see note that any relation R k as a subset of A k × A k +1 is r epresented by a one-direction bipartite digr aph D k . A ”complete relation” C k by definition is identi- fied with its one dire c tio n di-biclique gr aph d − B k . Any R k is a subset of C k . Corre sp ondingly one direction digraph D k is a subgraph of an one direction digraph of d − B k . The one dir e c tion digraph of d − B k is called s ince now o n the di- biclique i.e. by definition - a complete bipartite one direction di- graph. Another words: cobw eb p oset defining di- bicliques are links of a complete relations’ chain. According to the definition ab ove ar bitrary cobw eb pos e t Π = ( V , ≤ ) is a graded po set ( ranked p ose t) a nd for s ∈ N 0 : x ∈ Φ s − → r ( x ) = s, where r : Π → N 0 is a rank function on Π. Let us then define Kwa ´ sniewski finite cobw eb sub-po sets as follows Definition 1.2. L et P n = ( V n , ≤ ) , ( n ≥ 0) , for V n = [ 0 ≤ s ≤ n Φ s and ≤ b eing the induc e d p artial or der r elation on Π . Its easy to see that P n is ranked po set with ra nk function r as ab ove. P n has a unique minimal ele ment 0 = h 1 , 0 i ( with r (0) = 0). More over Π a nd all P n s satisfy the Jordan ch ain condition and the length of P n is l ( P n ) = r ( P n ) = n for n ≥ 0. F or finite graded p oset P one c a n define (see [1]) Whitney n umber s of the first and second kind w k ( P ) and W k ( P ) respe c tively as follows w k ( P ) = X x ∈ P , r ( x )= k µ (0 , x ) , 3 W k ( P ) = X x ∈ P , r ( x ) = k 1 = |{ x ∈ P : r ( x ) = k }| , where µ sta ys for M¨ obius function o f P indispensa ble in n umerous in version type formulas o f coun tless applications (see [1, 9, 11, 12, 13]). Then the characteristic polynomial of P [9, 1 1, 1 2, 13] is the p olynomia l χ P ( t ) = X x ∈ P µ (0 , x ) t n − r ( x ) = n X k =0 w k ( P ) t n − k , where n = l ( P ). Here next we answer the question p osed b y A.K.K wa ´ sniewski in the source pape r for the problem in question [7]. L et { P n } n ≥ 0 b e the se qu en c e of finite c obweb su bp osets (...). What is t he form and pr op erties o f { P n } n ≥ 0 ’s char acteristic p olynomials { ρ n ( λ ) } n ≥ 0 ? ( ...) Wh at ar e r e curr enc e r elations defining the family { ρ n ( λ ) } n ≥ 0 ? 2 Whitney n um b ers of cob w eb p osets Obviously for ar bitrary cobw eb pose t Π and for a ll its finite subp osets P n , ( n ≥ 0) one has: W k (Π) = F k , k ≥ 0 , (1) where { F n } n ≥ 0 is a natural n um bers v a lue d s equence uniquely designating Π. Now let us c onsider the corresp onding num b ers w k (Π). The explicite formula for M¨ obius function of the Fib onacci cob w eb p oset uniquely desig nated by the Fi- bo nacci se q uence was derived by the present author in [2, 3]. It can b e easy e x tend to the hole family o f cobw eb p osets and their finite subpo sets P n , ( n ≥ 0),[4]. Mor e - ov er , by the use o f notion of the standa r d reduced incidence algebra R (Π), (see [4 ]) one can show, that for x ∈ Π the v alue µ (0 , x ) dep ends on r ( x ) only . So for x as ab ov e we ha ve: µ (0 , x ) = µ ( r (0) , r ( x )) = µ (0 , r ( x )) = µ ( r ( x )) . (2) Moreov er µ (0 , x ) = µ ( r ( x )) = ( − 1 ) r ( x ) r ( x ) − 1 Y i =1 ( F i − 1) . (3) Then Prop ositi on 2.1. F or arbitr ary c obweb p oset Π and for al l its finite subp osets P n , ( n ≥ 0 ) c orr esp onding Whitney numb ers of the first kind ar e given by the formulas: 4 for k > 0 w k (Π) = X { x ∈ Π : r ( x )= k } µ (0 , x ) = F k · µ (0 , x ) = F k · ( − 1) k · k − 1 Y i =1 ( F i − 1) (4) and w 0 (Π) = 1 . (5) 3 The c haracteristic p olynomials of finite c obw eb p osets The k nowledge of Whitney n umber s w k ( P n ), enables us to constr uct the character- istic p olynomials for all P n , ( n ≥ 0). Let us recall the formu la defining χ n ( t ): χ P n ( t ) = X x ∈ P n µ (0 , x ) t n − r ( x ) = n X k =0 w k ( P n ) t n − k . Using the ab ov e for mu las one has Theorem 3.1. The char acteristic p olynomials χ P n ( t ) , ( n ≥ 0 ) ar e given by t he fol lowing explicit formula: χ P n ( t ) = χ n ( t ) = x n + n X k =1 ( − 1) k F k · k − 1 Y i =1 ( F i − 1) x n − k . (6) Moreov er, as in cas e of Fibonacci co bw eb pos e t, the following holds: Corollary 3 . 1. L et { F n } n ≥ b e the se quenc e designating the c obweb p oset Π (and al l c orr esp onding sub-p osets P n ). In the c ase F 1 = 1 ( or e quivalently | Φ 1 | = 1 ) one has χ n ( t ) = t n − t n − 1 (7) for n ≥ 1 and χ 0 ( t ) = 1 . (8) Corollary 3 . 2. L et { F n } n ≥ b e the se quenc e designating the c obweb p oset Π (and al l c orr esp onding s ub-p osets P n ). Th en the se quenc e { χ n ( t ) } n ≥ 0 of { P n } n ≥ 0 ’s char- acteristic p olynomials is define d by the following r e curr enc e r elation χ 0 ( t ) = 1 , χ 1 ( t ) = t − F 1 (9) χ n ( t ) = tχ n − 1 ( t ) + ( − 1) n F n ( F n − 1 − 1)( F n − 2 − 1) ... ( F 1 − 1) , n ≥ 2 . (10) 5 Example 3 .1. Let the sequence o f finite cobw eb p os ets { P n } n ≥ 0 be designated by the sequence { F n } n ≥ 0 such that F n = n + 1 (i.e. by the sequence N of natural nu m be rs). The examples of corresp onding characteristic p olynomia ls are: χ 0 ( t ) = 1 , χ 1 ( t ) = t − 2 , χ 2 ( t ) = t 2 − 2 t + 4 , χ 3 ( t ) = t 3 − 2 t 2 + 4 t − 18 , χ 4 ( t ) = t 4 − 2 t 3 + 4 t 2 − 18 t + 120 , χ 5 ( t ) = t 5 − 2 t 4 + 4 t 3 − 18 t 2 + 120 t − 105 0 , χ 5 ( t ) = t 6 − 2 t 5 + 4 t 4 − 18 t 3 + 120 t 2 − 1050 t + 1134 0 . Example 3 .2. Let the sequence o f finite cobw eb p os ets { P n } n ≥ 0 be designated by the sequence { F n } n ≥ 0 such that F 1 = 1 and F n = 2 n + 1 for n ≥ 1. The examples of corre s p o nding c ha racteristic polyno mia ls are: χ 0 ( t ) = 1 , χ 1 ( t ) = t − 3 , χ 2 ( t ) = t 2 − 3 t + 10 , χ 3 ( t ) = t 3 − 3 t 2 + 10 t − 56 , χ 4 ( t ) = t 4 − 3 t 3 + 10 t 2 − 56 t + 43 2 , χ 5 ( t ) = t 5 − 3 t 4 + 10 t 3 − 56 t 2 + 432 t − 422 4 , Example 3 .3. Let the sequence o f finite cobw eb p os ets { P n } n ≥ 0 be designated by the sequence { F n } n ≥ 0 such that F 1 = 1 a nd F n = k for n ≥ 1 and fo r so me k > 1. The examples of corresp onding characteristic polynomia ls are: χ 0 ( t ) = 1 , χ 1 ( t ) = t − k, χ 2 ( t ) = t 2 − kt + k ( k − 1) , χ 3 ( t ) = t 3 − kt 2 + k ( k − 1) t − k ( k − 1) 2 , χ 4 ( t ) = t 4 − kt 3 + k ( k − 1) t 2 − k ( k − 1) 2 t + k ( k − 1) 3 . In general one has χ n ( t ) = t n − kt n − 1 + k ( k − 1) t n − 2 + ... + ( − 1) n k ( k − 1) n − 1 , n ≥ 1 . Ac knowledgemen ts Discussions with Participant s of Gian-Carlo Ro ta Polish Seminar, ht tp://ii.uwb.edu.pl/akk/sem/sem rota.htm ar e highly appreciated. References [1] J oni S.A., Rota. 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