On Cobweb Admissible Sequences - The Production Theorem

On Cobweb Admissible Sequences The Pr o duction The or em Maciej Dziemia´ nczuk Student in the Institute of Computer S cience, Bia lystok Universit y (*) PL-15-887 Bi a lystok, st. Sosnow a 64, Poland e-mail: Maciek.Ciupa@gmail .com (*) f ormer W a rsa w Un ive rsit y Div ision Summary In this note fur ther clue decisiv e observ ations on cobw eb admiss ib le se- quences are shared with the audience. In particular an announced pro of of the Theorem 1 (by Dziemia´ nczuk) from [1] announced in In dia -Kolk ata- Decem b er 2007 is deliv ered h ere. Namely here and there we claim that any cob web admissible sequ ence F is at the p oint pro duct of primary co b w eb ad- missible sequences taking v al ues one and/or certain p ow er of an a ppropr iate primary n u m b er p . Here also an algorithm to pro duce the f amily of all cob w eb -admissible s e- quences i.e. the Pr oblem 1 from [1] i.e. one of sev eral prob lems p osed in source pap ers [2, 3] is solv ed us in g th e idea and metho ds implicitly p resen t already in [4]. Presen ted at Gian-Carlo P olish S eminar: http://ii.uwb.e du.pl/ akk/sem/sem r ota.htm 1 Preliminaries The notation from [2, 3, 1] is b eing here taken for gran ted. Definition 1 ( [2, 3, 1 ]) A se quenc e F is c al le d c obweb-admissible iff for any n , k ∈ N ∪ { 0 }  n k  F = n F · ( n − 1) F · ... · ( n − k + 1) F 1 F · 2 F · ... · k F ∈ N (1) Problem 1 ([2, 3, 1]) Find effe ctive char acterizations and/or an algorithm to pr o duc e the c obweb admissible se quenc es i.e. find al l examples. 1 2 Primary cob w eb admissible sequence Throughout this pap er w e s hall consequ en tly us e p letter only for p rimary n um b ers. Definition 2 A c obweb admissible se quenc e P ( p ) ≡ { n P } n ≥ 0 value d one and/or p owers of one c ertain primary numb er p i.e. n P ∈ { 1 , p, p 2 , p 3 , ... } is c al le d primary c obweb admissible se quenc e. Theorem 1 Any c obweb-admissible se quenc e F is at the p oint pr o duct of primary c obweb- admissible se quenc es P ( p ) . P R OOF Giv en any cob w eb admissible sequence F = { n F } n ≥ 0 , eac h of its elemen ts can b e represented as a pro duct of primary num b ers’ p o w ers i.e. n F = Q s ≥ 1 p α ( n,s ) s . Therefore the sequence F is at the p oin t pro d u ct of sequences P ( p 1 ) , P ( p 2 ) , ... suc h that P ( p s ) ≡ { n P s } n ≥ 0 and n P s = p α ( n,s ) s . Eac h of primary sequences P ( p s ) where s = 1 , 2 , 3 , ... is cob w eb admissible as follo wing h olds for any n, k ∈ N ∪ { 0 }  n k  P ( p 1 ) ·  n k  P ( p 2 ) · ... =  n k  F ∈ N ⇒ ⇒  n k  P ( p s ) = n k P ( p s ) k P ( p s ) ! = p N s p K s ∈ N (2) where N stands for the sum of index p o wers’ of primary num b ers p s in n F , ( n − 1) F , ..., ( n − k + 1) F pro du ct expan s ion via primary num b ers and corresp ondin gly K is th e ind ex p o wers’ su m for k fi r st elemen ts of th e se- quence F  3 Primary cob w eb admissible sequences family In this section w e define a family A ( p ) of all primary cob web admissible sequences ta king v alues one and/or certain p o wer of an appropriate p rimary n um b er p . In the next part of this section we presen t the family in th e graph structure of a tree defined in algorithmic wa y in w h at f ollo ws. F or this aim let consider a pr imary cob w eb adm issible sequence F ≡ P ( p ) and its corr esp onding family of sequences B ( F ) ≡ { n B ( F ) } n ≥ 0 suc h that n B ( F ) = m ↔ n F = p m In the sequel we shall consider sequences for arbitrary but fix ed one pr imary n um b er p , th erefore w e use abbreviatio P ( p ) ≡ P . 2 Lemma 1 Natur al numb er value d se quenc e F ≡ { n F } n ≥ 0 is primary c obweb admissible P ( p ) iff for any natur al numb er n , n F ∈ { 1 , p, p 2 , p 3 , ... } and ∀ 1 ≤ k ≤ ⌊ n/ 2 ⌋ n X s = n − k +1 s B ( F ) ≥ k X s =1 s B ( F ) . P R OOF The first steep . Giv en an y pr imary cob web admissible sequence F ≡ { n F } n ≥ 0 . F rom the Defin ition 2 w e kno w that n F ∈ { 1 , p, p 2 , ... } for certain primary n um b er p . F rom the Definition 1 we readily in fer that for an y n, k ∈ N ∪ { 0 }  n k  F = p n B ( F ) · ... · p ( n − k +1) B ( F ) p 1 B ( F ) · p 2 B ( F ) · ... · p k B ( F ) = p N p K ∈ N ⇒ N ≥ K where N = P n s = n − k +1 s B ( F ) and K = P k s =1 s B ( F ) . The second steep . Giv en an y sequence F ≡ { n F } n ≥ 0 where n F ∈ { 1 , p, p 2 , p 3 , ... } and ∀ 1 ≤ k ≤ ⌊ n/ 2 ⌋ P n s = n − k +1 s B ( F ) ≥ P k s =1 s B ( F ) (*). Then for an y natural n, k b elo w tak es place  n k  F = p n B ( F ) · ... · p ( n − k +1) B ( F ) p 1 B ( F ) · p 2 B ( F ) · ... · p k B ( F ) = C ∧ ( ∗ ) ⇒ C ∈ N  Definition 3 ( Primary cobw eb admissible tree) L et G ( p ) to b e a weighte d tr e e G ( p ) = h V , E , δ i wher e V stays for set of vertic es, E denotes a set of no des and fu nction δ which assigns weight for any vertex v ∈ V such that δ ( v ) ∈ { 0 , 1 , 2 , ... } and p - primary numb er. We shal l define the c orr esp onding g r aph G ( p ) via the fol lowing r e curr enc e: 1. v 0 ∈ V is called the ro ot with w eigh t δ ( v 0 ) = 0 2. If ( v 0 , v 1 , ..., v n − 1 ) is a path of grap h G ( p ) then ( v 0 , v 1 , ..., v n − 1 , v n ) is to o if, and only if ∀ 1 ≤ k ≤ ⌊ n/ 2 ⌋ N n,k ≥ K k ) where N n,k = P n i = n − k +1 δ ( v i ) and K k = P k i =1 δ ( v i ) . Conclusion 1 An y path ( v 0 , v 1 , ..., v n ) from the ro ot v 0 to v ertex v n enco des the first n terms with 0 F of prim ary cob web admissible sequence F ≡ { n F } ≥ 0 , n F ∈ { 1 , p, p 2 , ... } with help of elemen ts’ exp onent p o w ers’ sequence B ( P ) suc h that k B ( F ) = δ ( v k ) i.e. the n + 1-tuple ( v 0 , v 1 , ..., v n ) ∈ V n +1 ↔ ( δ ( v 0 ) , δ ( v 1 ) , ..., δ ( v n )) exactly enco des fi n ite primary cob we b ad m issible sequence F v alued by one and/or p o w ers of p rimary num b er p . 3 Observ ation 1 If any p ath ( v 0 , v 1 , ..., v n ) enc o des n the first terms with 0 F of prima ry c obweb adm issible se quenc e F then ther e exists infinite numb er of suc c essors vertic es v n +1 which enc o de primary c obweb admissible se que nc e F ′ sp e cifie d by these n first terms with 0 F and the one add itional ( n + 1) F ′ = δ ( v n +1 ) term. P R OOF If an y path ( v 0 , v 1 , ..., v n ) encodes n the first terms with 0 F of primary cob web admissible sequence F then there exists infi nite n um b er of n atural n um b ers M suc h that δ ( v n +1 ) = M and N n,k − 1 + M ≥ K k . Consequent ly , now we presen t an algorithm to generate primary cob w eb admissible tree. Algorithm 1 (primary cobw eb-admissible tree ) We sh al l b e gin with the r o ot v 0 of gr aph G ( p ) fr om D efinition 3 and i n the next ste eps, fr om any p ath ( v 0 , v 1 , ..., v n ) we obtain the very next one ( v 0 , v 1 , ..., v n , v n +1 ) . Input: An y path ( v 0 , v 1 , ..., v n ) of G ( p ) whic h enco des n the first terms with 0 F of primary cob web admissible sequence F . Output: Non-empt y s et ∅ 6 = ∆ n ⊆ { v n +1 : δ ( v n +1 ) ∈ { 0 , 1 , 2 , ... }} with v er tices’ successors for v n v er tex suc h that the p aths ( v 0 , v 1 , ..., v n , v n +1 ) where v n +1 ∈ ∆ n +1 enco des primary cob web-admissible sequence, to o. Under the con venien t notation for ve rtices v ( s ) ≡ v n +1 ∧ δ ( v n +1 ) = s n ote no w the f ollo wing. Steps: 1. If n = 1 ∆ 1 = { v (0) , v (1) , v (2) , ... } 2. If n = 2 ∆ 2 = { v ( m ) , v ( m + 1) , v ( m + 2) , ... } , w here m = δ ( v 1 ) ... n. F or an y natural n ∆ n = { v ( m ) , v ( m + 1) , v ( m + 2) , ... } , where m = max { K k − N n − 1 ,k − 1 : k = 1 , 2 , 3 , ..., ⌊ n/ 2 ⌋} where K k = P k i =1 δ ( v i ) an d N n,k = P n i = n − k +1 δ ( v i ). Definition 4 Denote with letter A ( p ) the f amily of al l primary c obweb ad- missible se quenc es P ( p ) . 4 Observ ation 2 The family A ( p ) is lab el le d-designate d b y the set of infinite p aths ( v 0 , v 1 , v 2 , ... ) of gr aph G ( p ) fr om the r o ot v 0 i.e. F ∈ A ( p ) ⇔ ( v 0 , v 1 , v 2 , ... ) is a p ath of gr aph G ( p ) wher e F ≡ { n F } n ≥ 0 and n F = p δ ( v n ) . P R OOF This is a conclusion on graph G ( p ) (Definition 3 ). The first steep . If giv en any p rimary cob w eb admissible sequence F ≡ { n F } n ≥ 0 , n ∈ { 1 , p, p 2 , ... } , then from the Definition 1 of admissibilit y and the Definition 3 of tree G ( p ) for an y natural num b ers n, k the f ollo wing is true  n k  F = p n B ( F ) · ... · p ( n − k +1) B ( F ) p 1 B ( F ) · p 2 B ( F ) · ... · p k B ( F ) = p N p K = p δ ( v n ) · ... · p δ ( v n − k +1 ) p δ ( v 1 · ... · p δ ( v k ) ∈ N where s B ( F ) = δ ( v s ) from Conclusion 1. In view of the Defin ition 1 N ≥ K hence ( v 0 , v 1 , v 2 , ... ) is a path of graph G ( p ). The second steep . T ak e any giv en p ath ( v 0 , v 1 , v 2 , ... ) of the grap h G ( p ). Then b y definition for an y natural num b er n , k , N n,k ≥ K k where N n,k = P n i = n − k +1 δ ( v i ) and K k = P k i =1 δ ( v i ). Hence this path does enco de the v ery primary cob web admissible sequence P ( p )  Theorem 2 ( Cob w e b Admissible Sequences Pro duction Theorem) The family of al l c obweb admissible se quenc es is a pr o duct of families A ( p s ) for s = 1 , 2 , 3 , ... i.e. for any c obweb admissible se que nc e F F ∈ × s =1 A ( p s ) P R OOF This is the summarizing conclusion. An y cob web admissib le sequen ce F is at the p oint pr o duct of primary cob w eb admissible sequen ces P ( p ) (Th eorem 1) and the family of a ll primary cobw eb admissib le sequences A ( p ) is defi n ed b y primary cob web admissible tr ee G ( p ) (Observ ation 2)  Ac knowledgemen ts I would lik e to thank Professor A. Kr zysztof Kwa ´ sniewski for his sup- ply of n ames for ob jects and op erations and fi nal improv emen ts of this p a- p er. Also discussions with Partic ipan ts of Gian-Carlo Rota Polish Seminar http://ii.uwb.e du.pl/ akk/sem/sem r ota.htm are appreciated. 5 References [1] A. Krzysztof Kwa ´ sniewski, M. Dziemia ´ nczuk, Co bweb p osets - R e- c ent R esults , ISRAMA 2007, Decem b er 1-17 2007 K olcata , INDIA, arXiv:0801 .3985, 25 Jan 2008 [2] A. Kr zysztof Kwa ´ sniewski, Cobweb p osets as nonc ommuta tive pr efabs , Adv. Stud. Con temp. Math. vol. 14 (1) (2007) 37-47. [3] A. Krzysztof Kwa ´ sniewski, On c obweb p osets and their c ombinatorial ly admissible se quenc es , arXiv:math.CO/05125 78 , 21 Oct 2007. [4] M. Dziemia´ nczuk, O n c obweb p osets tiling pr oblem , arXiv:math.Co/070 9.426 3, 4 Oct 2007 6