On inversion formulas and Fibonomial coefficients

On in v ersion form ulas and Fibono mial coeffi cien ts A. Krzysztof Kwa ´ sniewski member of the Institute of Combinato rics and its Applications Bialystok Universit y (*), F acult y of Physics PL - 15 - 424 Bialystok, ul. Lipow a 41, Polan d kw andr@gmail.com Ewa Krot-Sienia wsk a Bia lystok Unive rsity , Institut e of Computer Science PL - 15 - 887 Bialystok, ul. Sosno w a 64, Po land ew akrot@wp.pl (*) former: W arsa w Unive rsity Division FECS’08: The 2008 In ternational Conference on F rontiers in Education: W ORLDCOMP’08 Summary A researc h problem for und ergraduates an d graduates is b eing p osed as a cap for t he prior antece dent regular discrete mathematics ex ercises. [Here cap is not n ecessarily CAP=Competitive Access Pro vider, though nevertheless ...] The ob ject of t he cap problem of final interes t i.e. array of fi b onomial co efficients and the issue of its combi- natorial meaning is to be found in A.K.Kw a ´ sniewski’s source papers. The cap problem num ber seven - still op ened for students has b een placed on Mathemagics page of the first author [http://ii.u wb.edu.pl/akk/dydaktyk a/dy skr/dyskretna.htm ]. The indica- tory references are to p oint a t a part of the va st domain of t he foundations of computer science in ArXiv affiliation noted as CO.cs.DM. The presentatio n has b een verified in a tutor system of comm unication with a couple of intelligen t stud ents. The result is top secret.T emporarily . [Con tact: Wikip edia; Theory of cognitive d evel opment]. MCS num bers: 05A19 , 11B39, 15A09 Keywo rds: inv ersion formula s, fib onomial coefficients presented at the Gian-Carlo Rota Pol ish Seminar http://ii .uw b.edu.pl/akk/sem/sem rota.h tm 1. In the r ealm of knownnes s. Inv ersion form ulas. Ex.1 Pro ve that X k ≥ 0 n k ! k l ! ( − 1) k − l = δ nl (1) HINT: F or th at to do use th e follo wing ( x + 1) n = X k ≥ 0 n k ! x k ⇐ ⇒ x n = X k ≥ 0 n k ! ( x − 1) k ⇐ ⇒ ⇐ ⇒ x n = X k ≥ 0 X l ≥ 0 n k ! k l ! x l ( − 1) k − l ⇐ ⇒ (1) . Ex.2 Show t h at n X k =0 n n k o » k l – ( − 1) n − k = δ nl , n ≥ 0 (2) HINT: Use the com binatorial interpretation of S t irling num b ers’ of the I and the I I kind, ( ˆ n k ˜ and ˘ n k ¯ , respectively). Also note: x n = n X k =0 h n k i x k , x n = n X k =0 n n k o x k , x n = ( − 1) n ( − x ) n . Then x k = k X l =0  k l ff x l ∧ x l = ( − 1) i ( − x ) l , l ≥ 0 = ⇒ (2) . Ex.3 Pro v e that x n = n X k =0 n n k o ( − 1) n − k x k , x n = n X k =0 h n k i ( − 1) n − k x k , n ≥ 0 . HINT: U se Exercise 2. Ex.4 “ ` n k ´ q ” − 1 =? , for ` n k ´ q = n q ! k q !( n − k ) q ! , n q ≡ 1 − q n 1 − q , n, k ≤ 0. This problem is solv ed. Just contact pp.70&106 in [1 ] (in p olish) and note that ` n k ´ q counts ob jects from the L ( n, q ) lattice. Then one has 0 @ n k ! q 1 A − 1 = 0 @ n k ! q ( − 1) n − k q ( n − k 2 ) 1 A . Ex.5 Find the num b ers ( C n, k ) − 1 for the sequen ce C n, k b eing the uniq ue solution of the recurren ce relation, [3]: C n +1 , k = C n, k − 1 + 2 C n, k + C n, k +1 (3) C 0 , 0 = 1 , C k, 0 = 0 = C n, n + k , n, k > 0 . F or n, k > 0 one has ([3]): C n, k = ` 2 n n − k ´ k n . HINT: F rom th e ab ov e one h as: C − 1 0 , 0 = 1, C − 1 k, 0 = 0, C − 1 n, n + k = 0 for n, k > 0. Then one uses P k ≥ 0 ` 2 n k ´` k l ´ ( − 1) k − l = δ 2 n, l , to get X s ≤ n 2 n s ! n − s n s l ! n n − s ( − 1) s − l = δ nl , n, l > 0 . Ex.6 One can define the Charlier p olynomials , as orthogonal p olynomial sequence ( see [4 ], th e formula 1.13, a = − 1): P n ( x ) = 1 n ! n X k =0 n k ! x k . Let x n = X k ≥ 0 C n,k P k ( x ). Find thew num bers: C n,k and ( C n,k ) − 1 . HINT: U se Exercise 2 and x n = n X k =0 ˆ n k ˜ ( − 1) n − k x k . 2. Sp ecificit y b ey ond t he realm of knownnes s? Ex.7 Discover the in version formula i.e. the ar ray elements  n k  F  − 1 for  n k  F being the s o called fibonomia l coe fficie nts, i.e.  n k  F = n F ! k F !( n − k ) F ! , 2 for n F = F n being the n -th Fibona cci num ber, ( n, k > 0). [9,5,10,1 1,12] POSSIBLE SOLUTION: Let us consider the incidence alg ebra I (Π) of the Fi- bo nacci cobw eb po set Π and the s tandard reduced incidence a lgebra R (Π). These were reco gnized-discovered [Plato’s attitude ? ] in [5, 6, 7] and inv esti- gated there [L.E.J. Br ouw er constr uctivism attitude a nd co nstructivism: ht tp://en.wikip edia.or g/wiki/C o nstructivism (learning theory)] . Le t f : Π × Π → R be defined as fo llows f ( x, y ) = f ( k , n ) =     n k  F k ≤ n 0 k > n for x, y ∈ Π, s uch that the se gment [ x, y ] = { z ∈ Π : x ≤ z ≤ y } is of type ( k , n ), i.e. r ( x ) = k , r ( y ) = n . It is obvious that f ∈ R (Π) (and of course f ∈ I (Π)). Then for g = f − 1 being inv erse of f in R (Π) (also in I (Π)) o ne has following [7] ( f ∗ g )( k , n ) = X k ≤ l ≤ n F l g ( k , l ) f ( l , n ) = δ nk , i.e. X k ≤ l ≤ n F l  n l  F g ( k , l ) = δ nk . (4) Hence in o rder to discover the magic formula for  n k  F  − 1 one ha s to find out an explicit formula for g = f − 1 . Rig ht ? Maybe it can be r ecov ered using of the standard for mula fo r an inv erse element in I (Π) ?, (see for example [1, 8]). References [1] W. Lipski, W. Marek: Combinatoria l Analysis , v.59 B M, PWN, W arsaw 1986 (in polis h) [2] I. Bec k: Partial Or ders a nd the Fib onac ci N u mb ers , The Fibonacci Quaterly , 26 (1990) , pp.27 2 -274 . [3] L.W. Shapir o: A Catalan T riangle , Discrete Math. 14.1 (197 6 ), pp.83-90. [4] T.S. Chihara: An Intr o duction to Ort ho gonal Polynomials , Gor don& Breach, 1 978. [5] Krot E.: The first asc ent into the Fib onac ci Cobweb Poset , Adv anced Stud- ies in Contempora ry Mathematics 11 (2 005), No. 2, p.179 -184, ArXiv: math.CO/041 1007, cs.DM http://arxiv.or g /abs/ma th/0411007 [6] Krot- Sieniawsk a E .: On incidence a lgebras description o f co bw eb po sets, arXiv:080 2.370 3 , cs.DM http://arxiv.org /abs/0 802.3703 [7] E.Kro t-Sieniawsk a: R e duc e d Incidenc e algebr as description of c obweb p osets and KoDA Gs , ArXiv: 0802.42 93, cs.DM http://arxiv.o rg/a bs/080 2.4293 [8] Spiegel E., O’Donnell Ch.J.: In cidenc e algeb r as , Marcel Dekk er, Inc. Basel 1997 [9] Kwa ´ sniewski A.K .: T owar ds ψ -ext ension of Finite Op er ator Calculus of R ota Rep. Math. Phys. vol.48 , No3 (2001) pp.304- 342 cs.DM cs.NA ArXiv: math/0402 078 h ttp://arxiv.or g/abs/ math/0402078 3 [10] Kwa ´ sniewski A.K.: First observatio ns on Pr efab p osets‘ Whitney n umb ers , Adv ances in Applied Clifford Algebr as V olume 18, Num ber 1 / F ebruary , 2008, p. 57-7 3. O NLINE FIRST, Springer Link Date, August 10, 2007 , arXiv:080 2.169 6 , cs.DM http://arxiv.org /abs/0 802.1696 [11] Kwa ´ sniewski A.K.: On c obweb p osets and their c ombinatoria l ly admissible se qu en c es , ArXiv:math.Co/05 1257 8 v4 21 O ct 2 007, submitted to Gr aphs and Combinatorics; cs.DM http://arxiv.or g/abs/ math/0512578 [12] Kwa ´ sniewski A.K., Dziemia ´ nczuk M.: Cobweb p osets - R e c ent R esults , ISRAMA 20 07, Decem ber 1 -17 20 07 Kolk ata, INDIA, ar Xiv:0801 .3985 , cs.DM http ://ar xiv.org / abs/08 01.3985 4