A proof of Catalan s Convolution formula

Transcription

1 A proof of Catala s Covolutio formula Alo Regev Departmet of Mathematical Scieces Norther Illiois Uiveristy DeKalb, IL regev@mathiuedu arxiv: v1 [mathco] 2 Sep 2011 Abstract We give a ew proof of the k-fold covolutio of the Catala umbers This is doe by eumeratig a certai class of polygoal dissectios called k-i- dissectios Furthermore, we give a formula for the average umber of cycles i a triagulatio 1 Itroductio The Catala umbers are defied as follows Defiitio 1 For ay 0, For < 0, C 0 C 1 ( 2 +1 The Catala k-fold covolutio formula is due to Catala Theorem 2 [2] Let 1 k The i 1 ++i k C i1 1 C ik 1 k 2 k ( 2 k (1 Catala s origial proof [2, 3, 4, 5] uses Lagrage iversio Gessel ad Lacrombe [4] give two proofs which use hypergeometric idetities Tedford [6] exhibits several iterpretatios of the left-had side of (1 I this ote we use aother such iterpretatio, i terms of dissectios of polygos, to give a ew proof of Theorem 2 We arrive at this proof usig Theorem 5, which eumerates a class of polygoal dissectios called k-i- dissectios As aother coseuece of this eumeratio, i Corollary 7 we give a formula for the average umber of cycles i a triagulatio 2 The k-i- dissectios Defiitio 3 Let 3 ad let 0 k 3 1

2 Figure 1: Example of a 5-i-12 dissectio 1 A k-dissectio of a -go is a partitio of the -go ito k+1 parts by k ocrossig diagoals 2 A triagulatio of a -go is a ( 3-dissectio 3 For k 4, a k-i- dissectio is a ( k-dissectio of a -go ito oe k-go ad k+1 triagles (see Figure 1 A 3-i- dissectio is a triagulatio with oe of its 3 triagles marked 4 Let f k ( be the umber of k-i- dissectios It is well kow that for 3 the umber of triagulatios of a -go is C 2 Lemma 4 Let 3 k The k+1 ( kf k ( C i 1 f k ( i+1 (2 Proof The left-had side of (3 is the umber of k-i- dissectios, with oe of the k diagoals marked These ca also be chose as follows Choose oe vertex v out of the vertices, the choose 2 i k + 1 Form the diagoal from v to a vertex which is a distace i from v (proceedig, say, couterclockwise alog the edges of the -go Mark this diagoal Now choose a triagulatio of the resultig (i+1-go ad a k-i-(( i+1 dissectio of the resultig (( i + 1-go Each such choice results i a uiue k-i- dissectio with oe of the diagoals marked Lemma 4 ca be used to eumerate the k-i- dissectios Theorem 5 Let 3 k The umber of k-i- dissectios is ( 2 k 1 f k ( (3 2

3 Note6 There is a bijectiobetweek-i-dissectiosad k-crossigpartitios of {1,}, as defied i [1] Thus Theorem 5 is euivalet to [1, Theorem 1] Theorem 5 implies the followig corollary: Corollary 7 Let 3 k < The average umber of cycles of legth k i a triagulated -go is ( 2 k 1 Ck 2 C 2 Proof Each cycle of legth k i a triagulatio of a -go uiuely correspods to a k-i- dissectio together with a triagulatio of a k-go The result the follows from (3 The followig lemmas will be used i the proof of Theorem 5 It is well kow that for ay 0, C i C i C +1 (4 Lemma 8 For ay 1, Proof Note that Therefore by (4, ic i C i ic i C i i C i ( ic ic i C i 1 C i C i 2 2 C +1 ( 2+1 (5 ( 2+1 Lemma 9 Let 1 p 2 1 The ( p 1 2i C i ( p (6 Proof We use iductio o If 1 the p 1 ad both sides of 6 are eual to 1 Now suppose 2 If p the both sides are eual to 1 If p 2 1 the (6 follows from (4 ad (5, sice ( 2 2 2i C i C i ( ic 1 i C i C 1 i ( 2 1 C 2 ( 2 1 ic i C 1 i 3

4 Now suppose +1 p 2 2 Note that 1 p 1 ad p ( 1 1 Therefore by the iductio hypothesis, (6 holds for p 1 ad 1 Also p 1 ad p < 2 1, so that (6 holds for p 1 ad Thus ( p ( ( p 1 p ( ( 1 2i p 2 2i + i+1 i 2 i ( ( 1 2i p 1 2i i+1 i 1 i 1 i+1 ( 2i i ( p 2 2i 21 Proof of Theorem 3 Proof Fix k 3 ad proceed by iductio o If k the both sides are eual to 1 Now let k +1 By Lemma 4 ad by the iductio hypothesis, f k ( k 1 C i 1 f k ( i+1 k k 1 ( 2( i+1 k 1 C i 1 k i ( k i 1 C i 1 ( 2( i+1 k 1 i f k ( Solvig for f k ( ad applyig Lemma 9, with ad p 2 k, f k ( ( 2 k 2i 1 C i ( ( 2 k 2 k 1 2 k i 1 2 k 3 Proof of the Catala covolutio formula The ext Lemma gives the relatio betwee the umber of k-i- dissectios ad the Catala covolutio Lemma 10 Let 3 k < The kf k ( i 1 ++i k C i1 1 C ik 1 (7 4

5 Proof The left-had side of (7 is the umber of k-i- dissectios, with oe of the vertices of the k-go marked These ca also be chose as follows Choose ay vertex v of the -go For each vertex v, choose i 1,,i k such that i 1 ++i k This determies the legths of the sides of a k-go by startig at v ad proceedig, say, couterclockwise For example, i Figure 1, if v is the bottom vertex the the legths are 1,4,2,2,3 For each 1 r k, there is a resultig (i r +1-go sharig oe edge of the k-go Each of these (i r +1-go ca be triagulated i C ir 1 ways, formig a uiuely determied k-i- dissectio with oe of the of the k-go marked The proof of Theorem 2 ow follows from Lemma 10, sice i 1 ++i k C i1 1 C ik 1 k f k( k ( 2 k 1 k 2 k ( 2 k Refereces [1] M Bergerso ad A Miller ad A Pliml ad V Reier ad P Shearer ad D Stato ad N Switala, A ote o 1-crossig partitios, available at [2] E Catala, Sur les ombres de Seger, Red Circ Mat Palermo, 1 ( [3] D R Frech ad P J Larcombe, The Catala umber k-fold self-covolutio idetity: the origial formulatio, J Combi Math Combi Comput, 46 ( [4] I Gessel ad P J Lacrombe, A forgotte covolutio type idetity of Catala: two hypergeometric proofs, Util Math, 59 (2001, [5] P J Lacrombe, A forgotte covolutio type idetity of Catala, Util Math, 57(2000, [6] S J Tedford, Comibatorial iterpretatios of covolutios of the Catala umbers, Itegers 11 (2011 5