Inversion-descent polynomials for restricted permutations

Transcription

1 Inverson-descent polynomals for restrcted permutatons Fan Chung Ron Graham August 25, 2012 Abstract We derve generatng functons for a varety of dstrbutons of jont permutaton statstcs all of whch nvolve a bound on the maxmum drop sze of a permutaton π,.e., max{ π)}. Our man result treats the case for the jont dstrbuton of the number of nversons, the number of descents and the maxmum drop sze of permutatons on [n] {1, 2,..., n}. A specal case of ths gnorng the number of nversons) connects wth earler work of Claesson, Dukes and the authors on descent polynomals for permutatons wth bounded drop sze. In that paper, the desred numbers of permutatons were gven by samplng the coeffcents of certan polynomals Q k. We fnd a natural nterpretaton of all the coeffcents of the Q k n terms of a restrcted verson of Euleran numbers. 1 Introducton There s an extensve lterature on varous statstcs for S n, the set of all permutatons of {1, 2,..., n} e.g., see [1, 3, 4, 7, 8, 12, 13, 14, 16, 17, 18, 19, 20]). For a permutaton π n S n, we say that π has a drop at f π) <, and the drop sze s π). We say that π has a descent at f π + 1) < π). One of the earlest results [12] n permutaton statstcs asserts that the number of permutatons n S n wth k drops equals the number of permutatons wth k descents. Other statstcs for a permutaton π nclude the number of nversons of π.e., {, j) : < j, π) > πj)} ), and the major ndex of π.e, the sum of the ndces at whch a descent of π occurs). Many of these papers study the dstrbuton of the above statstcs and ther q-analogs as well as the dstrbuton of varous multvarate statstcs. In ths paper, we examne jont statstcs of permutatons wth the addtonal constrant on the maxmum drop sze. Enumeraton problems of permutatons wth bounded maxmum drop sze arse n the study of jugglng patterns as well as certan sortng algorthms. In [3], the descent polynomals wth bounded Unversty of Calforna, San Dego 1

2 maxmum drop sze were studed. In ths paper we extend the methods to examne the jont statstcs of nversons, descents and maxmum drop sze. The dervaton of the generatng functons of such combned statstcs of permutatons nvolves an nterplay of q-nomal coeffcents and varous modfed versons of Euleran numbers. An outlne of the paper s as follows. In Secton 2, we wll present our man result dealng wth the jont dstrbuton of descents and nversons over permutatons wth bounded drop sze. In Secton 3, we specalze ths result by gnorng nversons. Ths relates to earler work of Claesson, Dukes and the authors [3] on the same subject. In Secton 4, we wll show how to nterpret all the coeffcents n the polynomals arsng n [3] n terms of countng certan restrcted permutatons. Fnally, n Secton 5, we wll make some general comments and suggest a number of open problems. 2 Inversons, descents and maxdrop We begn by lstng some of the standard termnology we wll be usng. Wth [n] {1, 2,..., n}, we let S n denote the set of n! permutatons on [n]. We let DESπ) { [n] : π has a descent at } and we set desπ) DESπ). Also, we defne maxdropπ) max{ π)}. Fnally, we let nvπ) denote the number of nversons of π S n,.e., nvπ) { < j : π) > πj)}. For a formal parameter q, we use the standard defntons for Gaussan coeffcents: If we defne A nv,des n q, y) by [n] q q n 1, [n] q! [n] q [n 1] q [1] q, [ ] a [a] q! b [b] q![a b] q!, q Exp q z) q n 2) zn [n] q!. A nv,des n q, y) q nvπ) y desπ) π S n then a classc result of Stanley [17] shows that A nv,des n q, y) zn [n] q! 1 y Exp q zy 1)) y. 2.1) Our frst result can be thought of as a varant of 2.1) usng ordnary generatng functons rather than exponental generatng functons where we nclude a restrcton on the maxdrop of the permutatons as well. To state t, we frst 2

3 need several defntons. For a power seres P z) pn)zn, the notaton [z t ]P z) denotes the truncated sum n t pn)zn, whle [z t ]P z) denotes the sum n t pn)zn and [z t ]P z) denotes the sngle term pt)z t. We defne B n,k q, y) where S n,k {π S n : maxdropπ) k}. π S n,k q nvπ) y desπ) Theorem 2.1. For k 1, the generatng functon for B n,k satsfes B k B k q, y, z) B n,k q, y)z n F k G k, where [ ] k + 1 G k G k q, y, z) 1 q 2) j y 1) j 1 z j, j q j1 A nv,des A nv,des q, y, z) A nv,des n q, y)z n, F k F k q, y, z) [z k ]A nv,des G k ). Note that A nv,des s not the usual power seres of Stanley for nversons and descents. For example, for k 1, we have B 1 q, y, z) 1 qz q)z qy 1)z 2. Proof. We wll consder B n,k q, y + 1) q nvπ) y + 1) desπ) π S n,k IT, DESπ))q nvπ) y T π S n,k T [n] where For π S n, defne IT, S) { 1 f T S, 0 otherwse. tπ) max{ : π has descents at n, n + 1, n + 2,..., and n 1}, and defne tπ) 0 f πn 1) < πn). Thus, we have πn tπ)) > πn tπ) + 1) > > πn), 3

4 for tπ) > 0. Now, for π wth maxdropπ) k, we have πn) n k. Therefore, πn tπ) + j) n tπ) + j k for 0 j tπ). Hence, we can wrte B n,k q, y + 1) q nvπ) y + 1) desπ) π S n,k IS, DESπ))q nvπ) y S π S n,k S [n] 1 π S n,k T [n ] tπ) 1 IT, DESπ))q nvπ) y T ) For π S n, let π denote the reduced permutaton on [n tπ) 1]. That s, the mages πj), j [n tπ) 1], have the same relatve order as the mages πj), j [n tπ) 1], so the number of descents and nversons of π and π on ths nterval are the same). Note that for π S n, the number nversons occurrng at poston.e., the number of u < wth πu) > π)) s exactly n π) {j : πj) > π) for j > }. For example, for n, the number of nversons occurrng at poston n s just n πn). Contnung the proof, we have [z ]B k q, y + 1, z) B n,k q, y + 1)z n n n 1 π S n,k T [n ] tπ) 1 n 1 π S n,k T [n ] IT, DES π))q nvπ) y T + 1 z n IT, DES π))q nv π) y T z n ) a 1<a 2<...<a q j1 aj y 1 z ) where 0 a 1 < a 2 <... < a k are defned by the dentfcaton {πn tπ)), πn tπ) 1),..., πn)} {n a 1, n a 2,..., n a }. Thus, we have [z ]B k q, y + 1, z) B n,k q, y + 1)z n n 1 [z ] y 1 1 a 1<a 2<...<a q j1 aj y 1 z ) k ) ) B n q, y + 1, z) [z ] 1 + q j yz) 1 4

5 Comparng the coeffcents of z n for n k + 1, we can conclude that k [z ]B k q, y + 1, z) [z ] y 1 B k q, y + 1, z) 1 + q j yz) 1) ). Consequently, we have [z ] B k q, y + 1, z) 1 y 1 k 1 + q j yz) 1) )) 0 or, equvalently, [z ] B k q, y, z) 1 y 1) 1 k 1 + q j y 1)z) 1) )) 0 whch can be wrtten as ) [z ] B k q, y, z)g k q, y, z) 0 2.3) by choosng G k q, y, z) 1 y 1) 1 k ) 1 + q j y 1)z) ) We now set ) F k q, y, z) [z k ] B k q, y, z)g k q, y, z). 2.5) Snce F k only has powers of z at most k, we have [z k ]B k q, y, z) [z k ]A nv,des q, y, z) because permutatons on [n] cannot have drops of sze k or larger when n k. Thus, 2.5) can be wrtten as ) F k q, y, z) [z k ] A nv,des q, y, z)g k q, y, z). 2.6) From 2.3), we have B k q, y, z)g k q, y, z) F k q, y, z) and we conclude that B k F k. G k Fnally, we can transform G k nto the desred form usng the followng standard q-bnomal theorem e.g., see [9]): n q j t) n [ ] q 2) n t. q 0 5

6 Applyng 2.4) transforms G k to G k q, y, k) 1 y 1) 1 k ) 1 + q j y 1)z) 1 [ ] k q j 2) y 1) j 1 z j j q j1 as desred. Ths proves the theorem. Also, snce Let us now specalze Theorem 2.1 by settng y 1. In ths case we have Hn) G k y1 G k q, 1, z) 1 z[k + 1] q. A nv,des n q, y) y1 q nvπ) [n] q! π S n π S n q nvπ) 1 + q + q q n 1 )Hn 1) [n] q!. 2.7) Consequently, A nv,des G k ) y1 [n] q!z n) 1 [k + 1] q z). Ths mples that for 1 j k, the negatve of the coeffcent of z j s [k + 1] q [j 1] q! [j] q! [k + 1] q [j] q )[j 1] q! q j [k + 1 j] q [j 1] q!. Pluggng these expressons nto Theorem 2.1 wth y 1 yelds Corollary 2.2. The generatng functon for nversons and maxdrop s gven by H k z) π S n,k q nvπ) z n 1 k j1 qj [k + 1 j] q [j 1] q! z j. 2.8) 1 [k + 1] q z For example, for k 1, 2.8) yelds H 1 z) q nvπ) z n 1 qz q)z π S n, q) n z n+1 6

7 whch mples that for n 1, the number of π S n wth j nversons and maxdropπ) 1 s just ) n 1 j. We can gve an alternatve proof of Corollary 2.2 as follows. Let us thnk of the term q n the multpler below as beng assocated wth the choce of πn) wth n πn). For n > k, we can wrte H k n) π S n,k q nvπ) 1 + q + q q k )H k n 1) snce πn) n k) [k + 1] n k 1 q [k + 1] q!. Ths mples that the generatng functon H k z) for the H k n) s gven by H k z) H k n)z n [m] q! z m + [k + 1] q! z 1 [k + 1] q z m k whch proves 2.8). 1 k j1 qj [k + 1 j] q [j 1] q! z j 1 [k + 1] q z 3 Descents and maxdrop In ths secton, we specalze Theorem 2.1 by gnorng nversons. Ths n fact was the man focus of an earler paper of Claesson, Dukes and the authors [3]. We frst need a few defntons. We wll let n k denote the usual Euleran number [10]. It s a standard fact that n k enumerates the number of permutatons n Sn whch have k descents and also whch have k drops). The n th Euleran polynomal E n y) s defned by Thus, we have E n y) n k0 n y k y desπ). k π S n A nv,des q, y, z) q1 A nv,des 1, y, z) y desπ) z n E n y)z n. π S n On the other hand, ) k + 1 G k q, y, z) q1 G k 1, y, z) 1 y 1) j 1 z j j j1 y 1 + y 1)z). y 1 7

8 Consequently, ) ) k + 1 A nv,des G k ) q1 E n y)z n 1 )y 1) j 1 z j j j1 from whch t follows after a modest computaton) that π S n,k F k q1 [z k ]A nv,des G k ) q1 k t ) ) k E t y) y 1) j 1 E t 1 y) j y desπ) z n t1 j1 and so, we have the generatng functon also see [3]) 1 + k t1 E t y) t 1 j1 For example, when k 1, ths becomes π S n,1 y desπ) z n 1 z 1 2z y 1)z 2 j1 j j z t ) y 1) j 1 E t 1 y)) z t ) y 1) j 1 z j. 1 + z y)z y)z y + y 2 )z ) n y z n. 3.1) 2 0 Thus, f we let n denote the number of π S n whch have descents and maxdropπ) k, then n [1] s just the bnomal coeffcent n 2) there s a nce bjectve proof of ths fact that the reader may lke to fnd!). As t happens, Anders Claesson and Mark Dukes [5] earler had come across these permutatons n ther work on a class of sortng algorthms, and they notced that the same type of restrcted permutatons arose n the analyss of certan jugglng patterns [2]. In addton to seeng that n was just the [1] coeffcent of u 2 n the polynomal 1 + u) n, computaton suggested that n [2] was the coeffcent of u 3 n the polynomal 1 + u + 2u 2 + u 3 + u 4 )1 + u + u 2 ) n 2, and even further, that n [3] was the coeffcent of u4 n the polynomal 1 + u + 2u 2 + 4u 3 + 4u 4 + 4u 5 + 4u 6 + 2u 7 + u 8 + u 9 )1 + u + u 2 + u 3 ) n 3! Followng these clues, Claesson, Dukes and the authors [3] were able to confrm these conjectures wth the followng general theorem. 8

9 Theorem 3.1. Let n denote the number of π S n wth descents and maxdropπ) k. If n k, then n s equal to the coeffcent of u) n the polynomal P k u)1 + u u k ) n k where P k u) k E k j u )u 1) j k sj ) s u s. j For n k, n n the usual Euleran number) s equal to the coeffcent of u ) n the polynomal P k u). The frst few polynomals P k u) are gven n Table 1. Table 1: k P k u) u u + 2u 2 + u 3 + u u + 2u 2 + 4u 3 + 4u 4 + 4u 5 + 4u 6 + 2u 7 + u 8 + u u + 2u 2 + 4u 3 + 8u u u u u u u u u u u 14 + u 15 + u 16 In Table 1, we have ndcated n bold the coeffcents n the P k u) whch are guaranteed by the theorem to be Euleran numbers. However, we had no dea at that tme what the other coeffcents of P k u) mght mean, f anythng. Of course, snce they are postve ntegers, one could suspect that they dd have a nce nterpretaton. It turns out that ths suspcon was correct. Ths wll be the topc n the next secton. 4 Interpretng all the coeffcents of P k u). It wll be convenent to ntroduce the polynomals Q k u) u k P k u) for k 0. Thus, Q k u) s gven explctly as Q k u) k E k u )u 1) j 0 k s ) s u k s. We show the frst few Q k u) n Table 2. We show the same Q k u) as n Table 2 but ths tme wth the coeffcents arranged n a ) ) array C k. The, j) entry C k, j) of C k corresponds to the coeffcent of u )+j for 0, j k. Thus, we can wrte 9

10 Table 2: k Q k u) u + u 2 2 u 2 + u 3 + 2u 4 + u 5 + u 6 3 u 3 + u 4 + 2u 5 + 4u 6 + 4u 7 + 4u 8 + 4u 9 + 2u 10 + u 11 + u 12 4 u 4 + u 5 + 2u 6 + 4u 7 + 8u u u u u u u u u u u 18 + u 19 + u 20 Table 3: j j j j C C C 2 j C C 4 Q k u) C k, j)u )+j 0,j k We now ntroduce a stretched polynomal Qk u) defned by k Q k u) E k j u k+2 )u k+2 1) j k sj It follows see [3]) that Q k u) can also be wrtten as Q k u) C k, j)u k+2)+j+1 0,j k u s 10

11 Thus, Q k u) dffers from Q k u) n that 0 s are nserted n postons correspondng to u k+2), for 0 k + 1. For example, Q 0 u) u, Q 1 u) u 2 + u 4, Q 2 u) u 3 + u 5 + 2u 6 + u 7 + u 9, Q 3 u) u 4 + u 6 + 2u 7 + 4u 8 + 4u 9 + 4u u u 13 + u 14 + u 16. Representng the coeffcents of Q k u) n an array C k Ck, j) ), for 0 k and 0 j k + 1, we see that C k s formed from C k by addng an ntal column of 0 s as shown n Table 4. The followng key fact relatng Q to Q k Table 4: C C C 2 C C 4 was proved n [3]: Theorem 4.1 [3]). Q Q k 1 + u + + u ). 4.1) Note that the symmetry and unmodalty of the coeffcents of Q k u) and C k follow from ths result appled recursvely). In partcular, we have C k, j) C k k, k j) for 0, j k. As we noted earler, when n k, the condton that maxdropπ) k for π S n s automatcally satsfed. In ths case C k, k), the coeffcent of u )+k n Q k u) s just the Euleran number k k. Let us defne the restrcted Euleran number m j for 1 j m, to be the number of π S m wth desπ) and wth πm) j. It s clear for example 11

12 that m+1 m+1 m snce the only possble descents n any π Sm+1 wth πm + 1) m + 1 occur at places for 1 m 1. Thus, the entres C k, k) formng the rght-most column of C k can be replaced by the restrcted Euleran number. It turns out that all the entres of C k can be expressed as restrcted Euleran numbers. Theorem 4.2. For k 0, Q k u) 0,j k j+1 k + 1 u )+j. 4.2) Proof. We wll proceed by nducton on k, usng 4.1). For k 0, Equaton 4.2) certanly holds, snce Q 0 u) 1 and Check for k 1 f you are nervous about just usng the case k 0!). Assume that 4.2) holds for some k 0. Notce by 4.1) that each coeffcent of Q u) s a sum of k + 2 consecutve coeffcents of Qk u). However, each block of k + 2 consecutve coeffcents of Qk u) contans exactly one of the 0 s n the frst column of the correspondng array C k. Hence, each coeffcent of Q u) wll be a sum of k + 1 consecutve coeffcents of Q k u). There are two cases. a) The k + 2)-block of coeffcents of Q k u) starts wth one of the left-hand 0 s,.e., wth C k, 0). Then the sum of the entres s k + 1 C k, j) by the nducton hypothess. However, each π S k+2 wth desπ) +1 and πk +2) 1 corresponds to a unque π S wth desπ ) by defnng π t) πt) 1 for 1 t. The addtonal descent n π occurs at the place k + 1.) Thus, j1 j k + 1 j1 j 1 k whch s what we need. b) The k + 2)-block of coeffcents of Q k u) starts wth C k, r) for some r, where 1 r k + 1. Thus, the coeffcent sum s now j k + 1 r 1 j k jr However, we can argue as before that each π S k+2 wth desπ) k + 1 and πk + 2) r corresponds to a unque π S counted n one of the two sums. Namely, defne j+1 12

13 π s) { πs) f πs) < r, πs) 1 f πs) > r. It s easy to check that desπ ) desπ) f πk +1) < r and desπ ) desπ) 1 f πk + 1) > r. Ths mples that j k + 1 r 1 + jr j1 j k r k Snce these arguments hold for all 0 k, our nducton s complete, and the theorem s proved. We next deal wth the case when the maxdropπ) k condton comes nto play. Theorem 4.3. Q k u)1+u+u 2 + +u k ) n k 0 n 0 j k n + 1 n+1 k+j u )+j. 4.3) Proof. We wll proceed by nducton on n k. Equaton 4.3) holds for n k by Theorem 4.2. Suppose 4.3) holds for some n k where k 0 s fxed). The coeffcents of the powers of u n the product Q k u)1 + u u k ) are sums of k + 1 consecutve coeffcents of Q k u). Agan, these are two cases. a) The coeffcent sum s k n k+j n. In ths case, t s not hard to see that each π counted by n+1 counted by exactly one of the terms n k n k+j n k+j n n + 1 n+1 n the sum so that we have n+1. s n fact b) The coeffcent sum s k jr n k+j n r 1 + n k+j n, + 1 for some r, 1 r k. We clam ths sum s equal to n k+r n

14 As before, for π S n+1 wth desπ) + 1, πn + 1) n k + r and maxdropπ) k, we defne π S n by { π πs) f πs) < n k + r, s) πs) 1 f πs) > n k + r. It s easy to check that f πn) < n k+r then desπ ) desπ) and maxdropπ ) k, so that ths π s represented by the term n πn) +1 n the second sum. On the other hand, f πn) > n k + r then desπ ) desπ) 1 and maxdropπ) k, so that ths π s represented by the term n πn) n the frst sum. Snce these maps are nvertble then we have for all, 0 n, k jr n k+j n r 1 + n k+j n + 1 as desred. Ths completes the proof of Theorem Concludng remarks n k+dr n + 1, + 1 The fact that Q k u) and R k,n u) Q k u)1 + u u k ) n k are symmetrc rases some nterestng bjecton questons. For example, snce C k, j) C k k, k j), then we have j+1 k j+1 k + 1 k + 1, 0, j k. k Ths s not hard to see bjectvely by assocatng each permutaton π S n wth the unque permutaton σ S n gven by σt) k +2 πt), for t 1,..., k +1. More nterestng s the symmetry of R n,k u). Snce Q k u) has degree kk + 1) n u, then R n,k u) has degree kk + 1) + n k)k n + 1)k n u. The frst nonzero term of R n,k u) s u k whch has a coeffcent n+1 n+1 0 1). Also, the last nonzero term of R n,k u) s u n+1)k, whch also has a coeffcent 1. By Theorem 4.3, ths coeffcent of u n+1)k s n+1 n+1 k+j where n + 1)k k + 1) + j, 0 j k. That s, f j n + 1)k mod k + 1)), 0 j k, and n+1)k then n + 1 n+1 k+j 1. More generally, the coeffcents of u r and u r must be equal where r + r n + 2)k. Thus, for r k + 1) + j, r k + 1) + j, 0, j k and r + r n + 2)k, we have n + 1 n+1 k+j n n+1 k+j. 5.1)

15 For example, for k 2, n 4, r and r , we have [2] We lst the correspondng permutatons n Table 5. [2] Table 5: [2] [2] Is there an obvous bjecton whch proves 5.1)? Even for the smple case for the two sets of permutatons shown n Table 5, t s not clear what the correspondence should be! It seems to us that the maxdrop statstc can be combned wth other standard permutaton statstcs to produce nterestng results, e.g., such as n the recent paper of Hyatt and Remmel [11]. More generally, we beleve that there should be many smlar results for analogs to maxdrop such as the maxmum descent maxdes), the maxmum value of the number of nversons maxnv), the maxmum value of the major ndex maxmaj), etc., e.g., see [20]). These have not yet been explored but we hope to return to some of these questons n the future. Acknowledgements The authors wsh to acknowledge the very helpful comments of the referees n preparng the fnal verson of ths paper. References [1] D. Beck and J. B. Remmel, Permutaton enumeraton of the symmetrc group and the combnatorcs of symmetrc functons, J. Combn. Theory A ), [2] F. Chung and R. Graham, Prmtve jugglng sequences, Amer. Math. Monthly ), [3] F. Chung, A. Claesson, M. Dukes and R. Graham, Descent polynomals for permutatons wth bounded drop sze, European J. Combn., 31, 2010),

16 [4] R. J. Clarke, E. Stengrímsson and J. Zeng, New Euler-Mahonan statstcs on permutatons and words, Adv. n Appl. Math ), [5] A. Claesson and M. Dukes, personal communcaton. [6] L. Euler, Methodus unversals seres summand ulterus promota, Commentar academae scentarum mperals Petropoltanae ), Reprnted n hs Pera Omna, seres 1, volume 14, [7] D. Foata and G. Han, q-seres n Combnatorcs; permutaton statstcs Lecture Notes), prelmnary edton, [8] D. Foata and M.-P. Schützenberger, Major ndex and nverson number of permutatons, Math. Nachr ), [9] G. Gasper and M. Rahman, Basc Hypergeometrc Seres, Cambrdge Unv. Press, Cambrdge, England, 1990.) [10] R. L. Graham, D. E. Knuth and O. Patashnk, Concrete Mathematcs, Addson-Wesley, [11] M. Hyatt and J. Remmel The classfcaton of 231-avodng permutatons by descents and maxmum drop preprnt, 2012). [12] P. A. MacMahon, Combnatory Analyss, 2 volumes, Cambrdge Unversty Press, London, Reprnted by Chelsea, New York, [13] D. Rawlngs, Enumeraton of permutatons by descents, descents, major ndex, and basc components, J. Combn. Theory A ), [14] O. Rodrgues, Note sur les nversons, ou dérangements produts dans les permutatons, J. de Math ), [15] M. Skandera, An Euleran partner for nversons, Sém. Lothar. Combn /02), Art. B46d, 19 pp. electronc). [16] R. P. Stanley, Ordered structures and parttons, Memors Amer. Math. Soc ), 107 pp. [17] R. P. Stanley, Bnomal posets, Möbus nverson, and permutaton enumeraton, J. Combn. Theory A ), [18] R. Stanley, Enumeratve Combnatorcs, vol. 1, Wadsworth and Brooks/Cole,Pacfc Grove, CA, 2 nd prntng, 1996). [19] J. Shareshan and M. L. Wachs, q-euleran polynomals: excedance number and major ndex, Electron. Res. Announc. Amer. Math. Soc ), [20] J. Shareshan and M. L. Wachs, Euleran quassymmetrc functons, Adv. n Math ),