A symmetrical Eulerian identity

Transcription

1 Joural of Combiatorics Volume 17, Number 1, 29 38, 2010 A symmetrical Euleria idetity Fa Chug, Ro Graham ad Do Kuth We give three proofs for the followig symmetrical idetity ivolvig biomial coefficiets ( m) ad Euleria umbers m : a + b a 1 a + b b 1 for ay positive itegers a ad b (where we tae 0 0 0). We also show how this fits ito a family of similar (but more complicated) idetities for Euleria umbers. 1. Itroductio Euleria umbers, itroduced by Euler i 1736 [5], while ot as ubiquitous as the more familiar Beroulli umbers, Stirlig umbers, harmoic umbers, or biomial coefficiets, evertheless arise i a variety of cotexts i eumerative combiatorics, for example, i the eumeratio of permutatios with a give umber of descets [7]. Because the recurrece for Euleria umbers is a bit more complicated tha for may other families of special umbers, ad because they icrease i size rather rapidly, it was stated i [7] that, We do t expect the Euleria umbers to satisfy as may simple idetities. Nevertheless, the followig idetity is rather elegat ad appears to be ew. Theorem 1. For positive itegers a ad b, (1) a + b a 1 a + b. b 1 We poit out here that we will use the covetio that the Euleria umber 0 0 is 0 (istead of the more commo covetio i which this is tae to be 1). 29

2 30 Fa Chug et al. Eve for the special case of b 1, the resultig idetity is iterestig. It states that for ay positive iteger a, a +1 2 a+1 1. a 1 Equatio (1) loos superficially lie the well ow Worpitzy idetity [7, 9] x + (2) x 0 (which coverts betwee two bases for polyomials over C), but is actually quite differet. Oe differece beig that the ruig idex of the Euleria umber i (2) is o the bottom, whereas i (1) it is o the top. I this ote, we will give three differet proofs of Theorem 1 ad also derive some extesios of it. These idetities arose i a recet study of descets i permutatios which have a restrictio o their largest drop [4]. 2. A direct proof We start with the followig basic Euleria idetities [7]:, for 0; m m 1 m +1 (3) ( 1) (m +1 ), for >0. m 0 Lemma 1. For two positive itegers a ad b, a + b a 1 b 1 a + b +1 (b p) a+b p 1 (1 b + p) p+1 p +1 p 1 b 1 a + b (b p) a+b p 1 (1 b + p) p. p p0 Proof. a + b a 1 a + b a + b a + b a 1

3 A symmetrical Euleria idetity 31 ( ( )) b a + b +1 a + b a + b a + b +1 a + b +1 b 0 b a + b +1 a + b a + b +1 b 0 }{{} X ( ) b a + b a + b. a + b +1 b 0 }{{} Y We further expad X usig (3): X a + b +1 a + b a + b +1 b b a + b +1 b a + b +1 ( 1) j (b +1 j) a+b a + b +1 j 0 j0 b b a + b +1 + j ( 1) j (b +1 j) a+b + j j 0 j0 b 1 p+1 a + b +1 p +1 ( 1) j (b p) a+b p+j 1 p +1 j p 1 j0 ( b 1 p+1 a + b +1 )(b p) a+b p 1 p +1 ( 1) j (b p) j p +1 j p 1 j0 b 1 a + b +1 (b p) a+b p 1 (1 b + p) p+1. p +1 p 1 I a similar way, we have Y ( ) a + b a + b a + b +1 b ( ) b a + b b a + b +1 ( 1) j (b +1 j) a+b a + b +1 j 0 j0 b b a + b + j 1 ( 1) j (b +1 j) a+b + j 1 j 0 j0

4 32 Fa Chug et al. b a + b p p (b p) a+b p 1 ( 1) j (b p) j p j p0 j0 b 1 a + b (b p) a+b p 1 (1 b + p) p. p p0 Now we will use the followig biomial idetity of Abel [1]: For >0, α 0 ad β real, (4) (x + α) α 0 (x + β) (α β) 1. We use the otatio i Lemma 1 with a + b X Y. a 1 1 By usig (4), substitutig α a, β 1, a + b +1, a + b p, x 1+a, wehave b 1 a + b +1 X (b p) a+b p 1 (1 b + p) p+1 p +1 p 1 1 a a + b +1 a ( a) 1 (1 + a ) a+b +1. a + b +1 0 Also, maig the same substitutios as above but with a + b i (4), we have b 1 a + b Y (b p) a+b p 1 (1 b + p) p Together, we have X Y p0 p 1 a a 0 a + b ( a) 1 (1 + a ) a+b. a + b a a + b +1 ( 1) (a ) 1 (a +1) a+b +1 0

5 A symmetrical Euleria idetity 33 a a + b ( 1) (a ) 1 (a +1) a+b 0 a a + b +1 ((a ( 1) ) +(a ) 1) (a +1) a+b 0 a a + b ( 1) (a ) 1 (a +1) a+b 0 a a + b +1 ( 1) (a ) (a +1) a+b 0 a a + b + ( 1) (a ) 1 (a +1) a+b 1 1 a 1 a + b +1 ( 1) +1 (a 1) +1 (a ) a+b a 1 a + b ( 1) (a 1) (a ) a+b 1. 0 The above expressio is exactly equal to a + b b 1 0 by usig Lemma 1 agai but iterchagig a ad b. This completes the first proof of Theorem A bijective proof We first trasform the right-had side of (1) usig the reflectio property of Euleria umbers, ad settig a + b, to (5) a 1. a Now we iterpret the left-had side (LHS) as the umber of strigs of legth o the alphabet {1, 2,...,a, } such that each of the pairs (2, 1), (3, 2),..., (a, a 1) occurs as a ot-ecessarily-cosecutive substrig. For example, oe such strig whe 10ada 4 is A strig with o- symbols correspods to oe of the permutatios of elemets that

6 34 Fa Chug et al. are eumerated o the LHS; i this case we may regard it as a permutatio of {0, 1, 2, 3, 5, 6, 7, 9}, amely of the idices j i the strig x 0...x 1 where x j. This permutatio is supposed to be oe of the 3 that have exactly 3 descets; ideed, it is (First write dow the idices j that have x j 1, the write those with x j 2, etc.) The geeral case follows i the same way. The sum o the right-had side (RHS) will be ozero oly whe 0 a. Iterpretig it as above, the case 0 correspods to strigs of legth o {1, 2,...,a,a+1} that cotai (2, 1), (3, 2),...,ad(a +1,a). The case 1 is similar, but o the alphabet {1, 2,...,a, }. It cotais (2, 1), (3, 2),..., ad (a, a 1) ad it must have exactly oe. The case of geeral has alphabet {1,...,a+1, }, cotais(j +1,j) for 1 j< a +1, ad has exactly occurreces of. We ow costruct a bijectio betwee these two sets of strigs. LHS RHS: For a strig σ i the LHS, let deote the least idex (possibly 0) such that either ( +1, ) or(, ) appears i σ. Wemapσ to a strig τ i the RHS by the followig rule: 1 i if i<, leftmost other s 1 j j +1 ifj>. Note that sice (i, i) does t appear i σ for i<, but i does, the τ has exactly s. RHS LHS: Let τ be a strig i the RHS which has s. We first map these s to the elemets, 1,...,2, 1 i order, with the leftmost beig mapped to. The, all 1 s to the left of the leftmost get mapped to, ad all 1 s to the right of the leftmost get mapped to. Fially, for 2 i a +1, wemapito i + 1. (We recommed that the reader carry out these mappigs o a few specific examples to get a feelig for what is happeig! For example, i the example metioed previously with 10, a 4, we have ad ) To complete the proof, it is ow just a matter of checig that these two mappigs are ideed a bijectio betwee the LHS ad the RHS of (5) (which we leave to the reader) ad the proof is complete.

7 A symmetrical Euleria idetity A geeratig fuctio proof The geeratig fuctio for our modified Euleria umbers (i.e., with 0 0 0)is (6) E(w, z) ez e wz e wz we z,i w i z i!. (This is obtaied by subtractig 1 from the usual geeratig fuctio for the Euleria umbers, which is Eq. (7.56) i [7].) First, we compute e wz E(w, z) (wz) w i z! i!,i w z! i w i z (,i )! 1 z! i i w ( )!,,i w i z i!.,,i Next, we compute we z E(w, z) w But by (6) wehave w 1!,,i,,i z w i z! i!,i z! i w i 1 z 1 (,i )! z i w ( i 1 ) i 1 ( )! w i z!. (e wz we z )E(w, z) e z e wz (1 w )z.!

8 36 Fa Chug et al. Hece, by idetifyig coefficiets of w i z i these expressios, we obtai for >0, (7) i i 1 1, if i 0, 1, if i 0, 0, otherwise. By jugglig the variables i (7), (settig a+b ad i a while jugglig), we ca recover (1). The reaso this approach wored was because the multiplier (e wz we z ) was divisible by the deomiator e wz we z of E(w, z). We could carry out the same argumet with ay multiple of e wz we z, for example, e 2wz w 2 e 2z. I this case, the terms correspodig to the right-had side of (7) are (e wz + we z )(e z e wz ) e (w+1)z e 2wz + we 2z we (w+1)z (w +1) z! w (w +1) z.! 2 w z + w! 2 z! Expadig the correspodig sums ad extractig the coefficiet of w i z yields for, i 0, ( 2 ) i i i ( ) i 2 2, if i 1, 2, if i 1, 0, otherwise. More geerally, if we use the multiplier e rwz w r e rz for a positive iteger r, we obtai for, i 0, r i C r (, i)+ r ( ) i r r, if i r 1, r, if i r 1, 0, otherwise,

9 A symmetrical Euleria idetity 37 where r 1 r 1 C r (, i) j i+j 1 (r j) i j+1 j i+j (r j) i j. i j +1 i j j1 j1 5. Cocludig remars A umber of questios remai uresolved, some of which we metio here. 1. Ca bijective proofs be foud for some of the more geeral idetities we have described? 2. Are there iterestig idetities which would result from taig other multiples of e wz we z? 3. Are there q-aalogs to some of these idetities? For related wor, see [3, 6] ad[8], for example. 4. It is well ow that also couts the umber of permutatios π o [] which have drops, i.e., elemets i [] for which π(i) <i. With this iterpretatio, we ca replace by δp (), defied for a arbitrary poset (P, ) to be the umber of permutatios π : P P which have drops, which meas elemets x P such that π(x) x. With this iterpretatio, it is sometimes possible to exted results ivolvig Euleria umbers to this more geeral settig. For example, such a extesio is ow for the Worpitsy idetity. Theorem 2 ([2]). For a poset (P, ) o poits, ad ay positive iteger a, we have a + (8) δ P () χ G(P ) (a) where G(P ) is the icomparability graph geerated by (P, ) ad χ G(P ) is the chromatic polyomial of G(P ). Whe P [], liearly ordered by size, the G(P ) is the empty graph o vertices ad χ G(P ) (a) a, so that (8) reduces to the Worpitsy idetity (2). Is it possible to exted our results i this directio? Refereces [1] N. H. Abel, Beweis eies Ausdrucs, vo welchem die Biomial-Formel ei eizeler Fall ist, Joural für die reie ud agewadte Mathemati 1 (1826), Frech traslatio i Abel s Œuvres Compléte, secod editio, 1 (1881),

10 38 Fa Chug et al. [2] J. Buhler ad R. L. Graham, A Note o the Biomial Drop Polyomial of a Poset, J. Comb. Th (A) 66 (1994), MR [3] L. Carlitz, A ote o q-euleria umbers, J. Comb. Th (A), 25 (1978), MR [4] F. Chug, A. Claesso, M. Dues ad R. L. Graham, Eumeratig permutatios with specified umbers of drops ad descets, Europea J. Comb., to appear. [5] L. Eulero, Methodus uiversalis series summadi ulterius promota, Commetarii acdemiae scietiarum imperialis Petropolitaae 8 (1736), Reprited i his Opera Omia, series 1, volume 14, [6] H. O. Foules, A orecusive combiatorial rule for Euleria umbers, J. Comb. Th (A) 22 (1977), MR [7] R. L. Graham, D. E. Kuth ad O. Patashi, Cocrete Mathematics, Addiso-Wesley, secod editio, MR [8] D. H. Lehmer, Geeralized euleria umbers, J. Comb. Th (A) 32 (1982), MR [9] J. Worpitzy, Studie über die Beroullische ud Eulersche Zahle, Joural für die reie ud agewadte Mathemati 94 (1883), Fa Chug Uiversity of Califoria, Sa Diego Ro Graham Uiversity of Califoria, Sa Diego Do Kuth Staford Uiversity Received October 27, 2009