NOTE AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS

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1 NOTE AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS BERNARDO M. ÁBREGO, SILVIA FERNÁNDEZ-MERCHANT, AND BERNARDO LLANO Abstrct. Gven ntegers nd n, there s unque wy of wrtng n s n = n n... n so tht n < <n <n. Usng ths representton, the Krusl- Mculy functon of n s defned s m (n) = n n... n. We show tht f nd <m (n), thenm ()m (n ) m (n). As corollry, we obtn short proof of Mculy s Theorem.. Introducton Gven ntegers nd n, there s unque wy of wrtng n s n n n n =... (.) 2 so tht n <n 2 < <n <n. Usng ths representton, clled the -bnoml representton of n, thekrusl-mculy functon of n s defned s, n n n m (n) = (See [2], [5], [8], [] for detls.) The mn gol of ths note s to prove the followng nequlty for Krusl-Mculy functons nd show some of ts consequences. Theorem. Let,, nd n be ntegers such tht nd n. If<m (n), then m ()m (n ) m (n). (.2) Krusl-Mculy functons re relevnt, mong other thngs, for ther pplctons to the study of ntchns n multsets (see for exmple [8], [2]), posets, rngs nd polyhedrl combntorcs (see [4] nd the survey [3]). In prtculr, they ply nd mportnt role n provng results, extensons nd generlztons of clsscl problems concernng the Krusl-Kton ([9],[7]) nd Erdős-Ko-Rdo [6] theorems. For nstnce, s corollry of Theorem, we obtn short proof of Mculy s Theorem [] (see Secton 2). More recently, the uthors [] ppled Theorem to the problem of fndng the mxmum number of trnslted copes of pttern tht cn occur mong n ponts n d-dmensonl spce, typcl problem relted to the study of repeted ptterns n Combntorl Geometry. Theorem s tght n the sense tht (.2) does not necessrly hold f m (n). For nstnce, whenever 2, n 3 =4,n 2 =2,n =, nd = m (n), we he tht m () m (n ) =m (n) <m (n). We conclude ths ntroducton by pontng out tht the nterestng problem of chrcterzng the equlty cse n Theorem s stll open. Dte: November 2, Mthemtcs Subect Clssfcton. [2]Prmry 5A5; Secondry 5A2. Key words nd phrses. bnoml representton of postve nteger, Krusl-Mculy functon, shdow of set. Ths pper s n fnl form nd no verson of t wll be submtted for publcton elsewhere.

2 2 BERNARDO M. ÁBREGO, SILVIA FERNÁNDEZ-MERCHANT, AND BERNARDO LLANO 2. Consequences of the theorem Consder the set of nonncresng sequences of length M = (x,x 2,...,x ) N : x x 2... x ª wth ts lexcogrphc order. Tht s, for x nd y n M,wewrtex y f for some ndex, x <y nd x = y whenever <.IfA M,thentheshdow of A, denotedby A, conssts of ll subsequences of length of elements of A ( ( ) = ). Tht s, A = {x M : y A such tht x s subsequence of y}. Usng these defnton, observe tht f A, B M, then (A B) = A B. (2.) There s n mportnt reltonshp between shdows of sets nd Krusl-Mculy functons. Nmely,fwedenotebyF (N) the set consstng of the frst N members of M n the lexcogrphc order, then F (N) = m (N). (2.2) We now prove Mculy s Theorem s corollry of Theorem. Corollry. (Mculy s Theorem, 927 []) Let. For every A M, A F ( A ). Proof. Let A M. We proceed by nducton on A. If =or A =, the result s trvlly true. Suppose nd A 6=. Set A = {x M : x =nd x A}, A 2 = {x M : x 2 nd x A}, nda 2 = {x A : x 2}. Herex denotes the conctenton of x nd, tht s x s the -tuple x wth n entry ppended n the ( ) th poston. Clerly, A =(A ) (A 2 ) A 2 nd the terms n the unon re prwse dsont. Moreover, we cn ssume tht A A 2 6=. Otherwse, snce ll entres of members of A re 2, we cn wor wth the set A obtned by subtrctng to every entry n the sequences of A ( A = A nd A = A.) Let = A A 2 nd b = A 2.Notetht A = b nd. If x =(x,x 2,...,x ) A,then(x,x 2,...,x ) A nd (x,x 2,...,x, ) = x A. Thts,A A. Wenowclculte A n terms of A,A 2,ndA 2 usng (2.). A = A 2 A 2 A ( A ) ( A 2 ) = A 2 A 2 ( A ) ( A 2 ) = ( A 2 A 2 ) ( (A A 2 ) ). If x ( A 2 A 2 ),thenx 2. Thus( A 2 A 2 ) ( (A A 2 ) ) =, nd consequently A = A 2 A 2 (A A 2 ). (2.3) We consder two cses. If m ( A ), then A = A 2 A 2 (A A 2 ) A 2 A = m ( A ). Assume <m ( A ). Snce then b< A nd thus, by nducton nd (2.2), A 2 A 2 A 2 F (b) = m (b) nd (A A 2 ) F () = m (). Therefore, by (2.3), Theorem, nd (2.2); we he A m (b)m () m ( A ) = F ( A ). In terms of shdows of sets, nd usng our prevous corollry, Theorem cn be generlzed s follows.

3 AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS 3 Corollry 2. Gven sets A M nd B M wth A < F ( A B ) we he A B F ( A B ). Proof. By the prevous corollry nd (2.2), A B m ( A ) m ( B ) nd A < m ( A B ). Thus, by Theorem, m ( A )m ( B ) F ( A B ). 3. Proof of the theorem Frst n observton. If n> then by Pscl s dentty n n n 2 n n = (3.) Let = P short f =. = be the -bnoml representton of. Wesytht s long f, nd Lemm. Let be n nteger. m ( )=m (). If s short, then m ( ) = m (),otherwse Proof. The result s cler for =. If s short, then = P =v for some v 2 nd v v. Thus = P =v v s the -bnoml representton of. Then m ( )=m () v = m (). Now suppose s long. There s v 2 such tht = for <v, nd ether v = or v nd v > v. Then v 2 = v v 2 nd by (3.) the bnoml representton of s v =. v v Then, gn by (3.), µ v 2 v 3 m ( ) m () = =. v 2 v 2 To prove the Theorem, we need to consder the extended -bnoml representton of postve nteger, byrequrngn coeffcent. Tht s, we wrte =... 2, 2 wth = < < <. The condton = s necessry to me ths representton unque when t exsts. (Clerly =does not he n extended representton.) In fct, Lemm 2. Let = P =v be the -bnoml representton of, wherethetermsequl to zero he been omtted. The extended -bnoml representton of exsts (nd t s unque), f nd only f v v.

4 4 BERNARDO M. ÁBREGO, SILVIA FERNÁNDEZ-MERCHANT, AND BERNARDO LLANO Proof. If v v, then, by (3.), v v = v v v. Thus = P v v P = P =v s n extended -representton of. Recproclly, f = = s n extended -representton, then =, nd there s v such tht = for v wth ether v = or v > v. Then, by (3.), vx = = v, v = =v =v s the -representton of. Thus v = v v. We cn defne m () =P = for the extended -representton of (f t exsts). It turns outthtbothdefntons gree,.e., m () =m (). Indeed, f = P =v wth v, then by (3.) nd the lst proof, m () m () = v v =. X Let n = = n,= = P v v = X, nd n = b = b, be bnoml representtons. Lemm 3. If <m (n), then <n b. Proof. We prove the contrpostves. If n,then n = n m m (n), snce n n nd m s non-decresng functon by Lemm. Now, f b n, thenb< b n.thus n n n n n = n b>n =..., but n n n n m (n) =..., 2 nd clerly n n. Thus m (n). ProofofTheorem.Recll b = n. Clerly, (.2) holds f =,ndthecse =s consequence of Lemm. We frst prove the cse <b. Assume tht the pr (, b) mnmzes m ()m (b) wth s smll s possble. If, the result holds. Let = P =v 2. Suppose frst tht v v.thenhsnextendedrepresentton,sy = P = mn(,b ) mx(,b ) α = = nd β = = b =..Let Note tht b = α β nd α<. Also mn(,b ) < mn( 2,b 2 ) < < mn(,b ) nd < mx(,b ) < < mx(,b ) <b (snce <b by ssumpton). Therefore the defntons wegeforα nd β re -bnoml representtons (extended for β). Ths mens tht m (α)m (β) =m (α)m (β) =m ()m (b). Ths contrdcts the mnmlty of. s the - Assume now tht v = v. Ths mens tht = v P v = =v representton of, nd thus s short. Then by Lemm, m ( ) m (b )= m () m (b ) m ()m (b), whch gn contrdcts the mnmlty of. Now, ssume b.snce<m (n) then, by Lemm 3, <n b.tht s, = b = n. We proceed by nducton on. If =, then = b 2 = n 2.

5 AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS 5 Thus n 2 2 n = n = b = 2 b,.e., b = n. Hence, m ()m 2 (b) = n = m2 (n). Assume 2 nd tht the result holds for. Letn = n n,b = b n, nd = n. Clerly, b = n,nd <m (n ) snce <m (n) = n m (n ). By nducton on the result holds for,b,n,ndthus m (b)m () m (n) = n m (b ) n m ( ) n m (n ) = m (b )m ( ) m (n ). References [] B. M. Ábrego, S. Fernández-Merchnt nd B. Llno, On the mxmum number of trnsltes (submtted). [2] I. Anderson, Combntorcs of fnte sets. Dover Publctons, Mneol, NY, (22), ISBN , 45 pp. (Corrected reprnt of the 989 edton publshed by Oxford Unversty Press, Oxford.) [3] S. L. Bezruov nd U. Lec, Mculy Posets (Dynmc Survey), Electron.J.ofCombn.(25) #DS2, 4 pp. [4] L. J. Bller nd A. Borner, Fce numbers of polytopes nd complexes, In: Hndboo of Dscrete nd Computtonl Geometry, J. E. Goodmn nd J. O Roure, eds., CRC Press Ser. Dscrete Mth., CRC, Boc Rton, FL, 997, [5] G. F. Clements nd B. Lndström, A generlzton of combntorl theorem of Mculy. J. Combntorl Theory 7 (969) [6] P. Erdős, C. Ko, R. Rdo, Intersecton theorems for systems of fnte sets. Qurt. J. Mth. Oxford (2), 2, (96) [7] G. O. H. Kton, A theorem of fnte sets, In: Theory of grphs. (Proc. Colloq., Thny, 966) Acdemc Press, New Yor (968) [8] D. E. Knuth, The Art of Computer Progrmmng, Vol. 4, Fsc 3. Genertng ll combntons nd prttons. Addson-Wesley, Upper Sddle Rver, NJ, 25, v5pp. ISBN [9] J. B. Krusl, The number of smplces n complex. In: Mthemtcl Optmzton Technques (ed. R. Bellmn) Unv. of Clforn Press, Bereley (963) [] F. S. Mculy, Some propertes of enumerton n theory of modulr systems. Proc. London Mth. Soc., 26 (927) (B. M. Ábrego nd S. Fernández-Merchnt) Deprtment of Mthemtcs, Clforn Stte Unversty, Northrdge t 8 Nordhoff Street, Northrdge, CA 933 E-ml ddress: bernrdo.brego@csun.edu nd slv.fernndez@csun.edu Current ddress: (B. M. Ábrego nd S. Fernández-Merchnt) Centro de Investgcón en Mtemátcs, A.C., Gunuto, Gto., Mexco (B. Llno) Deprtmento de Mtemátcs, Unversdd Autónom Metropoltn, Iztplp, Sn Rfel Atlxco 86, Colon Vcentn, 934, Méxco, D.F. E-ml ddress: llno@xnum.um.mx