The Möbius function of partitions with restricted block sizes

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1 The Möbius fuctio of partitios with restricted block sizes Richard EHRENBORG ad Margaret A. READDY Departmet of Mathematics, Uiversity of Ketucky, Lexigto, KY USA Received 3 February 2006; accepted 30 August 2006 Abstract The purpose of this paper is to compute the Möbius fuctio of filters i the partitio lattice formed by restrictig to partitios by type. The Möbius fuctio is determied i terms of the descet set statistics o permutatios ad the Möbius fuctio of filters i the lattice of iteger compositios. Whe the uderlyig iteger partitio is a kapsack partitio, the Möbius fuctio o iteger compositios is determied by a topological argumet. I this proof the permutahedro makes a cameo appearace. MSC: 05A17; 05A18; 06A07 Keywords: Euler ad taget umbers; descet set statistic; set partitio lattice; r-divisible partitio lattice; kapsack partitios; permutahedro 1 Itroductio Sylvester [9] iitiated the study of subposets of the partitio lattice. He proved that the Möbius fuctio of the poset of set partitios where each block has eve cardiality is give by every other Euler umber, also kow as the taget umbers. Recall the Euler umbers eumerate alteratig permutatios. Staley [6] exteded this result by cosiderig set partitios where each block has cardiality divisible by r. I this case the Möbius fuctio is give by the umber of permutatios with descet set {r, 2r, 3r,...}. I this paper we cotiue to explore the coectio betwee partitio lattices ad permutatio statistics. To each set partitio we ca assig a type, which is the iteger partitio cosistig of the multiset of cardialities of the blocks. Give a set F of iteger partitios it is the atural to ask for the Möbius fuctio of the associated poset of set partitios whose types belog to F. We slightly modify this by istead workig with poited set partitios ad poited iteger partitios ad lettig F be a filter i the poset of poited iteger partitios. Partially supported by Natioal Sciece Foudatio grat

2 Figure 1: The poset I 4 of poited partitios of the iteger 4. I Theorem 3.1 we cosider poited set partitios whose types belog to give filter F of poited iteger partitios. The questio of computig the Möbius fuctio is reduced to the descet set statistics ad Möbius fuctios i the smaller ad more tractable lattice of iteger compositios. I Sectio 4 we cosider kapsack partitios, a otio motivated by a well-kow cryptosystem. Theorem 4.4 allows us to determie the Möbius fuctio of the iteger compositio lattice. We obtai explicit expressios for the sought-after Möbius fuctio i Theorem Poited iteger partitios, set partitios ad compositios Recall that a iteger partitio λ = {λ 1,..., λ k } of a o-egative iteger is a multiset of positive itegers havig sum, that is, = λ λ k. Defiitio 2.1 Let be a o-egative iteger. A poited iteger partitio of is a pair {λ, m} = {λ 1,..., λ k, m} where m is a o-egative iteger ad λ = {λ 1,..., λ k } is a iteger partitio of m. The iteger m is called the poited part. It is uderlied to distiguish it from the other parts of the partitio. Deote by I the collectio of all poited iteger partitios of the o-egative iteger. Partially order the set I by the two cover relatios ad {λ 1,..., λ k 2, λ k 1, λ k, m} {λ 1,..., λ k 2, λ k 1 + λ k, m} {λ 1,..., λ k 1, λ k, m} {λ 1,..., λ k 1, λ k + m}. I words, by addig two parts together we go up i the order. If oe of the parts is the poited part the the sum becomes the poited part i the ew poited partitio. Observe that the poset I is ot a lattice for 3. Defiitio 2.2 A poited set partitio π = (σ, Z) of a fiite set S cosists of a subset Z of S ad a partitio σ of the set differece S Z. 2

3 We call the subset Z the zero block of the poited set partitio π. Moreover, we deote the umber of blocks (icludig the zero block) of π by π. Let Π be the poset of all poited set partitios o the set {1,..., }, where the partial order is give by refiemet. That is, for two poited set partitios π ad π, we have that π π if every block of π is cotaied i some block (possibly the zero block) of π ad the zero block of π is cotaied i the zero block of π. Observe that Π is isomorphic to the partitio lattice Π +1 by isertig the elemet + 1 ito the zero block ad lettig the zero block be a ordiary block of the partitio. We defie the type of a poited partitio (σ, Z) to be the poited iteger partitio {λ, m} where m is the cardiality of the zero block Z ad λ is the multiset of the sizes of the blocks of σ, that is, λ = { B : B σ}. Defiitio 2.3 Let be a o-egative iteger. A poited iteger compositio of is a list c = (c 1,..., c k 1, c k ) of o-egative itegers with sum where c 1 through c k 1 are required to be positive. Note that the last etry c k is allowed to be 0, so we uderlie it to distiguish it from the other etries. Let C be the collectio of all poited compositios of. Partially order the elemets of C by the cover relatios (c 1,..., c j 1, c j, c j+1, c j+2,..., c k 1, c k ) (c 1,..., c j 1, c j + c j+1, c j+2,..., c k 1, c k ), (c 1,..., c k 2, c k 1, c k ) (c 1,..., c k 2, c k 1 + c k ). That is, the cover relatio occurs by addig two adjacet etries of the compositio. Observe that the poset C is isomorphic to the Boolea algebra o elemets. For a permutatio τ i the symmetric group o elemets S, the descet set is a subset of {1,..., 1} defied as {i : τ(i) > τ(i + 1)}. Give a permutatio τ i S with descet set {s 1, s 2,..., s k 1 } where s 1 < s 2 < < s k 1, defie the descet compositio of τ to be (s 1, s 2 s 1,..., s k 1 s k 2, s k 1 ). For a compositio c = (c 1,..., c k ) of let β( c) deote the umber of permutatios i the symmetric group S with descet compositio c, that is, with the descet set {c 1, c 1 + c 2,..., c c k 1 }. For a poited compositio c = (c 1,..., c k 1, c k ) with c k strictly greater tha 0, let β( c) be as i the o-poited compositio case. If the poited compositio c = (c 1,..., c k 1, c k ) satisfies k 2 ad c k = 0, let β( c) = 0. Lastly, for c = (0) let β( c) = 1. A uiform way to view the descet compositio statistic idepedetly of the last part is the followig. For a poited compositio c = (c 1,..., c k 1, c k ) let d be the compositio where the last part is icremeted by oe, that is, d = (c 1,..., c k 1, c k + 1). The β( c) is the umber of permutatios τ i S +1 havig descet compositio d ad satisfyig τ( + 1) = + 1. The type of a iteger compositio (c 1,..., c k 1, c k ) is the poited iteger partitio {c 1,..., c k 1, c k }. Observe that the last part of the compositio becomes the poited part. We remark that the three posets I, Π, ad C are graded, ad hece raked. Throughout we will deote the rak fuctio of a graded poset by ρ ad use ρ(x, y) to deote the rak differece ρ(x, y) = ρ(y) ρ(x). 3

4 3 The Möbius fuctio of restricted partitios Recall that a filter F (also kow as a upper order ideal) i a poset Q is a subset of Q such that if x y ad x belogs to F the y belogs to F. For S a subset of the poset Q, the filter geerated by S is give by {y Q : x S such that x y}. Note that if f is a order preservig map from a poset P to a poset Q ad F is a filter of Q the the iverse image f 1 (F ) is a filter of P. For further iformatio about posets, we refer the reader to Staley s treatise [7]. Let F be a filter i the poited iteger partitio poset I. Let Π (F ) be the filter of the poited set partitio lattice Π cosistig of all set partitios havig their types belogig to F, that is, Π (F ) = {π Π : type(π) F }. Similarly, defie C (F ) to be the filter of poited compositios havig types belogig to F, that is, C (F ) = { c C : type( c) F }. Observe that both Π (F ) ad C (F ) are joi semi-lattices. Hece after adjoiig a miimal elemet ˆ0 we have that both Π (F ) {ˆ0} ad C (F ) {ˆ0} are lattices. Theorem 3.1 Let F be a filter of the poited iteger partitio poset I. The the Möbius fuctio of the filter Π (F ) with a miimal elemet ˆ0 adjoied is give by ( ) µ Π (F ) {ˆ0} = ( 1) ρ( c,ˆ1) µ C (F ) {ˆ0} (ˆ0, c) β( c). (3.1) c C (F ) Although C (F ) {ˆ0} is ot ecessarily graded, the expressio ρ( c, ˆ1) appearig i the statemet of Theorem 3.1 is well-defied sice the iterval [ c, ˆ1] is itself graded. Proof of Theorem 3.1: For a poited compositio c = (c 1,..., c k 1, c k ) of a o-egative iteger recall that the multiomial coefficiet ( ) c is defied as!/(c1! c k!). Observe that ( c) couts the umber of permutatios i S havig descet set cotaied i the set {c 1, c 1 + c 2,..., c c k 1 }. By the priciple of iclusio ad exclusio, we have that β( c) = c d ( 1)ρ( c, d) () d. Thus the right-had side of (3.1) is give by ( 1) ρ( c,ˆ1) µ C (F ) {ˆ0} (ˆ0, c) β( c) c C (F ) = = c C (F ) c d ( 1) ρ( d,ˆ1) µ C (F ) {ˆ0} (ˆ0, c) d C (F ) ( 1) ρ( d,ˆ1) = d C (F ) ( 1) ρ( d,ˆ1) ( ) d 4 ( d ˆ0< c d ). ( ) d µ C (F ) {ˆ0} (ˆ0, c)

5 The last sum ca be viewed as follows. A ordered set partitio is a partitio i Π with a orderig of the blocks such that the last block is the zero block. The type of a ordered set partitio is the poited compositio where oe lists the size of each block i their give order. Note that give a compositio d of, there are ( d ) ordered set partitios with d as its type. Hece we have that d C (F ) ( 1) ρ( d,ˆ1) ( ) d = = π ordered set partitio type(π) C (F ) π Π (F ) = π ( 1) ( 1) π ( π 1)! π Π (F ) µ Π (F ) {ˆ0} (π, ˆ1) = µ(π (F ) {ˆ0}). Example 3.2 We have the followig idetity coectig the Stirlig umbers of the secod kid with the Euleria umbers: ( ) k k j ( 1) j 1 (j 1)! S( + 1, j) = ( 1) k A(, j). (3.2) k j=1 Recall that the Stirlig umber of the secod kid S(, j) couts the umber of set partitios of a -elemet set ito j parts, whereas the Euleria umber A(, j) couts the umber of permutatios i S with j 1 descets. To prove equatio (3.2) let F be the filter of I cosistig of all poited iteger partitios with at most k parts. The the filter Π (F ) cosists of all poited set partitios with at most k parts. Usig the Stirlig umbers of the secod kid, we ca write the Möbius fuctio of Π (F ) {ˆ0} to be the left-had side of equatio (3.2). Similarly, the filter C (F ) cosists of all poited compositios of with at most k parts. Let c be a compositio with j parts. The the iterval [ˆ0, c] i the poset C (F ) {ˆ0} is a rak-selected Boolea algebra, more specifically, the Boolea algebra B j+1 with raks 1 through k removed. Hece the Möbius fuctio µ C (F ) {ˆ0} (ˆ0, c) is give by ( 1) k j+1 ( j k). Summig over all poited compositios c cosistig of j parts i the right-had side i Theorem 3.1, we obtai ( 1) k ( j k) times the umber of permutatios i S with j 1 descets. Hece the idetity follows. Whe k = this idetity states that µ(π +1 ) = ( 1)! sice Π (F ) {ˆ0} = Π = Π +1. j=1 We have the followig corollary. This result was proved with differet techiques i [5]. Corollary 3.3 Let = r p + m. Let Π,r,m cardiality at least m ad the remaiig blocks have cardiality divisible by r. fuctio of Π,r,m {ˆ0} is give by be all the partitios i Π where the zero block has The the Möbius ( ) µ Π,r,m {ˆ0} = ( 1) p+1 β(r, r,..., r, m). }{{} p 5

6 Proof: Let F be the filter of I geerated by the poited partitio {r, r,..., r, m}. The poset Π,r,m is exactly the filter Π (F ). Observe that C (F ) {ˆ0} has a uique atom, amely the poited compositio (r, r,..., r, m). Thus the Möbius fuctio µ C (F ) {ˆ0} (ˆ0, c) is o-zero oly for this compositio. Hece the summatio i the right-had side of (3.1) has oly oe term, amely ( 1) p+1 β(r, r,..., r, m). By settig m = r 1 i the previous corollary ad usig the bijectio betwee Π ad Π +1, we obtai the followig corollary due to Staley [6, 8]. Corollary 3.4 (Staley) Let = r p ad let Π r deoted the r-divisible lattice, that is, all partitios o elemets where the cardiality of each block is divisible by r ad a miimal elemet ˆ0 is adjoied. The the Möbius fuctio of Π r, µ(π r ), is give by the sig ( 1) p times the umber of permutatios τ i S with descet set {r, 2r,..., r} ad τ() =. 4 Kapsack partitios Let λ = {e m 1 1, em 2 2,..., emq q } be a partitio, that is, a multiset of positive itegers, where m i deotes the multiplicity of the elemet e i. We tacitly assume that all the e i s are distict, that is, e i e j for i j. Sice there are q i=1 (m i + 1) multi-subsets µ of λ, the followig iequality holds: q e : µ λ (m i + 1). (4.1) e µ Whe equality holds i (4.1), we call λ a kapsack partitio. Observe this is equivalet to that each iteger i the set o the left-had side of (4.1) has a uique represetatio as a sum of elemets from the multiset λ. Whe all the etries i λ are distict, that is, λ is a set, this defiitio reduces to the usual otio of a kapsack system appearig i cryptography. Example 4.1 (a) If e 1,..., e q, m 1,..., m q are positive itegers satisfyig the iequality j 1 i=1 m i e i e j for all j = 2,..., q the {e m 1 1,..., emq q } is a kapsack partitio. (b) If {λ 1,..., λ p } is a kapsack partitio, q a prime greater tha the sum λ 1 + +λ p ad j a positive iteger less tha q, the {j λ 1 mod q,..., j λ p mod q} is a kapsack partitio. i=1 A poited iteger partitio {λ, m} is called a poited kapsack partitio if λ is a kapsack partitio. I this sectio we cosider the filter F geerated by a poited kapsack partitio {λ, m}. We determie the Möbius fuctio of the poset C (F ) {ˆ0} ad thus obtai a explicit formula for µ(π (F ) {ˆ0}). Before proceedig oe more defiitio is eeded. Let V (λ, m) = V be the collectio of all poited compositios c = (c 1,..., c k 1, m) i the filter C (F ) such that whe each c i, 1 i k 1, is expressed as a sum of parts of λ, the summads for each c i are distict. Example 4.2 For the poited kapsack partitio {1, 1, 1, 4, m} the set V of poited compositios is V = {(1, 1, 1, 4, m), (1, 1, 5, m), (1, 1, 4, 1, m), (1, 5, 1, m), (1, 4, 1, 1, m), (5, 1, 1, m), (4, 1, 1, 1, m)}. 6

7 Observe the compositio (2, 1, 4, m) does ot belog to V sice 2 is the sum of two equal parts. Example 4.3 For the poited kapsack partitio {r, r,..., r, m} the set V oly cosists of the poited compositio (r, r,..., r, m). The ordered partitio lattice Q p cosists of all ordered partitios of the set {1,..., p} together with a miimal elemet ˆ0 adjoied. The cover relatio i Q p is to merge two adjacet blocks. The ordered partitio lattice is isomorphic to the face lattice of the (p 1)-dimesioal permutahedro; see for istace [1] or Exercise 2.9 i [2]. Hece Q p is Euleria ad has Möbius fuctio give by µ Qp (x, y) = ( 1) ρ(x,y). Theorem 4.4 Let F be the filter i the poited iteger partitio poset I geerated by the poited kapsack partitio {λ, m} = {λ 1,..., λ p, m} of the iteger. Let c = (c 1,..., c k 1, c k ) be a poited compositio i the filter C (F ). The the Möbius fuctio µ C (F ) {ˆ0} (ˆ0, c) is give by µ(ˆ0, c) = { ( 1) p k if c V, 0 otherwise. Proof: Cosider the lattice C (F ) {ˆ0} ad let c be a poited compositio i C (F ). Assume that the poited part of the compositio c is greater tha m. The iterval [ˆ0, c] i C (F ) {ˆ0} is itself a lattice ad each atom i this iterval has poited part equal to m. The joi of these atoms also has poited part equal to m, so the joi caot ot equal the compositio c. By Corollary i [7], the Möbius fuctio vaishes, that is, µ(ˆ0, c) = 0. Let P be the ideal of C (F ) cosistig of all poited compositios with poited part m. Observe that P has the maximal elemet (λ λ p, m). Hece P is a fiite joi semi-lattice ad thus P {ˆ0} is lattice. We first cosider the case whe all the parts of λ are distict. I this case P = V. There is a isomorphism betwee the lattice P {ˆ0} ad the ordered partitio lattice Q p. The isomorphism f seds the ordered partitio (B 1, B 2,..., B k ) to the poited compositio f((b 1, B 2,..., B k )) = λ i, λ i,..., λ i, m, i B 1 i B 2 i B k ad f(ˆ0) = ˆ0. Hece we coclude that the Möbius fuctio is give by µ(ˆ0, c) = ( 1) p k. For the geeral case we allow the partitio to cotai multiple etries. The ideal P is the isomorphic to a filter of the ordered partitio lattice Q p. Namely, let R be the collectio of all ordered partitios (B 1, B 2,..., B k ) such that if λ i = λ j for 1 i < j p the the elemets i ad j either appear i the same block or the elemet i appears i a block before the block cotaiig the elemet j. It is straightforward to verify that R is a filter of Q p ad that the map f is agai a isomorphism, this time from R to P. 7

8 Figure 2: The complex Γ correspodig to the poited kapsack partitio {1, 1, 1, 4, m}. I this figure the poited part m has bee omitted. Observe the faces ot o the boudary of Γ correspod to the set V. Refer to Example Recall that the boudary of the dual of the permutahedro is a simplicial complex p. We view this simplicial complex as the (p 1)-dimesioal sphere i R p cut by the ( p 2) hyperplaes xi = x j for 1 i < j p. The geometric picture is the sphere S p 1 with ( p 2) great spheres o it. Let H be the hyperplae arragemet i R p give by the collectio of the hyperplaes x i = x j whe λ i = λ j. Let C be the chamber C = {(x 1,..., x p ) : x i x j if i < j ad λ i = λ j }. Note that C is a coe i R p. The filter R i the poset Q p correspods to the subcomplex Γ of the complex p cosistig of all faces G i p that are cotaied i the chamber C. The geometric realizatio of Γ is the itersectio of the uit sphere S p 1 ad the chamber C, ad hece is homeomorphic to a (p 1)-dimesioal ball. Let L(Γ) deote the face lattice of Γ. A face G is o the boudary of Γ if ad oly if G is cotaied i oe of the hyperplaes i H. I other words, G is o the boudary of Γ if ad oly if the correspodig poited compositio c = (c 1,..., c k 1, m) has a etry c i, i < k, such that whe c i is expressed uiquely as a sum of parts of λ, two terms are equal. I this case the poited compositio c does ot belog to the set V. For a compositio c let G be the associated face i Γ. The we have that 0 if G is o the boudary of Γ, µ(ˆ0, c) = µ L(Γ) (G, ˆ1) = χ(γ) = 0 if G is the empty face, ( 1) ρ(g,ˆ1) otherwise, where the last step is Propositio i [7], provig the theorem. Combiig Theorems 3.1 ad 4.4, we have: Theorem 4.5 Let F be the filter i the poited iteger partitio poset I geerated by the poited kapsack partitio {λ, m} = {λ 1,..., λ p, m} of the iteger. The the Möbius fuctio µ(π (F ) {ˆ0}) of the filter Π (F ) with a miimal elemet ˆ0 adjoied is give by µ(π (F ) {ˆ0}) = ( 1) p 1 β( c), that is, ( 1) p 1 times the umber of permutatios i S whose descet compositio belogs to the set V. c V Observe that Corollary 3.3 is also a cosequece of Theorem 4.5 usig Example 4.3. I this case the complex Γ is a simplex. 8

9 Cotiuatio of Example 4.2 Let F be the filter i I geerated by the poited kapsack partitio {1, 1, 1, 4, m}. The ( ) µ Π (F ) {ˆ0} = β(1, 1, 1, 4, m) + β(1, 1, 5, m) + β(1, 1, 4, 1, m) +β(1, 5, 1, m) + β(1, 4, 1, 1, m) + β(5, 1, 1, m) +β(4, 1, 1, 1, m). 5 Cocludig remarks The poset Π (F ) {ˆ0} raises may atural questios. Whe the miimal elemets of the filter F have the same rak, the poset Π (F ) {ˆ0} is graded. Oe may ask if this is a shellable poset. Similarly, for a geeral filter, that is, whe the miimal elemets of the filter F have differet raks, the previous questio exteds to determiig if the order complex of Π (F ) {ˆ0} is o-pure shellable [3]. I the case whe the poset is ot shellable, ca oe still determie the homology groups of the order complex? Oe goal here is to obtai a bijective proof of Theorem 4.5. The symmetric group S acts o the order complex of Π (F ) {ˆ0}. This actio is iherited by the homology groups. Ca this represetatio be determied? Whe oe cosiders a poited kapsack partitio it is atural to cojecture that the poset is ideed is shellable ad hece the homology is cocetrated i the top homology. Furthermore, the dimesio of the top homology is give by the Möbius fuctio ad the actio of the symmetric group is give by the direct sum of Specht modules correspodig to the skewpartitios associated with the compositios i the set V. For the poited kapsack partitio {r, r,..., r, r 1} correspodig to the r-divisible lattice (recall Corollary 3.4), this research program has bee carried out. See the papers [4] ad [10] ad the refereces therei. Fially, a eumerative questio is to determie the umber of kapsack partitios of. This umber seems related to the prime factor decompositio of. Techiques from aalytic umber theory may be required. Ackowledgmets We thak the referee for may helpful commets ad the MIT Mathematics Departmet, where this paper was completed durig the authors sabbatical year. Refereces [1] L. J. Billera ad A. Saragaraja, The combiatorics of permutatio polytopes, i Algebraic Combiatorics (L. Billera, C. Greee, R. Simio ad R. Staley, eds.), DIMACS Series i Discrete Mathematics ad Theoretical Computer Sciece, America Mathematical Society, Providece, RI, [2] A. Björer, M. Las Vergas, B. Sturmfels, N. White, G. Ziegler, Orieted matroids, Cambridge Uiversity Press, Cambridge, [3] A. Björer ad M. Wachs, Shellable opure complexes ad posets. I, Tras. Amer. Math. Soc. 348 (1996)

10 [4] A. R. Calderbak, P. Halo ad R. W. Robiso, Partitios ito eve ad odd block size ad some uusual characters of the symmetric groups, Proc. Lodo Math. Soc. (3) 53 (1986) [5] R. Ehreborg ad M. Readdy, Expoetial Dowlig structures, preprit [6] R. P. Staley, Expoetial structures, Stud. Appl. Math. 59 (1978) [7] R. P. Staley, Eumerative Combiatorics, Vol. I, Wadsworth ad Brooks/Cole, Pacific Grove, [8] R. P. Staley, Eumerative Combiatorics, Vol. II, Cambridge Uiversity Press, [9] G. S. Sylvester, Cotiuous-Spi Isig Ferromagets, Doctoral dissertatio, Massachusetts Istitute of Techology, [10] M. L. Wachs, A basis for the homology of the d-divisible partitio lattice, Adv. Math. 117 (1996)