Order Ideals in Weak Subposets of Young s Lattice and Associated Unimodality Conjectures

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1 Annals of Cobinatorics 8 (2004) /04/ DOI ********************** c Birkhäuser Verlag, Basel, 2004 Annals of Cobinatorics Order Ideals in Weak Subposets of Young s Lattice and Associated Uniodality Conjectures L. Lapointe 1 and J. Morse 2 1 Instituto de Mateática y Física, Universidad de Talca, Casilla 747, Talca, Chile lapointe@inst-at.utalca.cl 2 Departent of Matheatics, University of Miai, Coral Gables, FL 33124, USA orsej@ath.iai.edu Received July 17, 2003 AMS Subject Classification: 06A06, 05A17, 05A10, 05E05 Abstract. The k-young lattice Y k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k > 0. The Y k poset was introduced in connection with generalized Schur functions and later shown to be isoorphic to the weak order on the uotient of the affine syetric group S k+1 by a axial parabolic subgroup. We prove a nuber of properties for Y k including that the covering relation is preserved when eleents are translated by rectangular partitions with hook-length k. We highlight the order ideal generated by an n rectangular shape. This order ideal, L k (, n), reduces to L(, n) for large k, and we prove it is isoorphic to the induced subposet of L(, n) whose vertex set is restricted to eleents with no ore than k + 1 parts saller than. We provide explicit forulas for the nuber of eleents and the rank-generating function of L k (, n). We conclude with uniodality conjectures involving -binoial coefficients and discuss how iplications connect to recent work on sieved -binoial coefficients. Keywords: Young lattice, uniodality 1. Introduction The Young lattice Y is the poset of integer partitions given by inclusion of diagras. This poset can be induced fro the branching rules of the syetric group, and certain order ideals of Y are in theselves interesting posets. For exaple, the induced subposet of partitions whose Ferrers diagras fit inside an n rectangle satisfies any beautiful properties. These principal order ideals, denoted L(, n), are graded, selfdual, and strongly sperner lattices [8]. Further, it is known that the nuber of eleents Research supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) project # , the Prograa Foras Cuadráticas of the Universidad de Talca, and by NSERC grant # Research supported in part by NSF grant #DMS

2 2 L. Lapointe and J. Morse p i (, n) of rank i in L(, n) are coefficients in the generalized Gaussian polynoial, and thus for a uniodal seuence [6, 10]. That is, [ ] n + p i (, n) i = = (1 n+1 ) (1 +n ) i 0 (1 ) (1. (1.1) ) Letting 1, the total nuber of eleents in this poset is given by ( ) n + L(, n) =. (1.2) A weak subposet Y k of the Young lattice was introduced in connection with functions that generalize the Schur functions [2, 3]. This poset (hereafter called the k-young lattice) is a lattice defined on the set of partitions whose first part is no larger than a fixed integer k 1. The order arises fro a degree preserving involution on the set of k-bounded partitions P k that generalizes partition conjugation. The involution sends one k-bounded partition λ to another, λ ω k, giving rise to a partial order on P k as follows: For λ and µ differing by one box, Young order on partitions: λ < µ when λ µ (and euivalently λ µ ). k-order on k-bounded partitions: λ µ when λ µ and λ ω k µ ω k. It happens that λ ω k = λ for large k iplying that the k-order is the Young order in the liit k. The k-young lattice originated fro a conjectured forula for ultiplying k-schur functions [2,3] that is analogous to the Pieri rule. In particular, the conjecture states that the k-schur functions s (k) µ appearing in the expansion of the product s 1 s (k) λ are exactly those indexed by the successors of λ in the k-young lattice. That is, s 1 s (k) λ = s (k) µ. λ µ In [5], it is shown that the k-young lattice is in fact isoorphic to the weak order on the uotient of the affine syetric group S k+1 by a axial parabolic subgroup and that the paths in Y k can be enuerated by certain k-tableaux, or by reduced words for affine perutations. Here we investigate general properties of the k-young lattice. Most notably, we reveal that partitions with rectangular shape and hook-length k, called k-rectangles, play a fundaental role in the structure of this poset. We prove for any k-rectangle and λ, µ P k, λ µ if and only if (λ ) (µ ) leading to the stronger stateent that: λ µ µ = µ and λ µ for soe k-bounded partition µ. This is a central property needed to identify the k- Young lattice with a cone in the perutahedron-tiling of the k-diensional space [11].

3 Order Ideals in Sublattices of Young s Lattice 3 The significance of k-rectangles also plays an iportant role at the syetric function level in that ultiplying a Schur function indexed by a k-rectangle with a k-schur function is trivial [4]. That is, for any k-bounded partition λ, s s (k) λ = s(k) λ. Following our study of the k-rectangles and other properties of the k-young lattice, we discuss a faily of induced subposets of L(, n) whose vertex set consists of the eleents that fit inside an n rectangle and have no ore than k +1 parts strictly saller than. Surprisingly, we find that these subposets are isoorphic to the principal order ideal of Y k generated by the shape n. As such, we denote these order ideals by L k (, n) and note that they are graded, self-dual, distributive lattices of rank n. We provide explicit forulas for the nuber of vertices and the rank generating function: λ = λ L k (,n) [ ] k k+1 1 (n k+ 1) 1 [ ] k, (1.3) 1 for n k + 1, which iplies that the coefficient of i in the right hand side of this expression is the nuber of partitions in L k (, n) with rank i. Further, letting 1, ( ) ( ) L k k + 1 k (, n) = + (n k + 1), (1.4) 1 for n k + 1. Since the vertex set of L k (, n) is contained in that of L k+1 (, n), these order ideals provide a natural seuence of subposets of L(, n). That is, L(, n) can be constructed fro the chain of partitions with no ore than one row saller than by successively adding sets of partitions with exactly j parts saller than for j 2. This decoposition aids our investigation of uestions pertaining to uniodality. Propted by the uniodality of L(, n), we coputed exaples that suggest L k (, n) is uniodal in certain cases. In particular when k 1 od p for all prie divisor p of. When is prie, we find that the uniodality of L k (, n) relies on the conjecture: Conjecture 1.1. If k 1, 0 od, then the coefficients of the powers in (1 (n k+) [ ] ) k 1 (1, (1.5) ) for a uniodal seuence for all n k + 1. We also generalize this conjecture to include the case when is not prie (see Conjecture 8.2). We conclude with a discussion of how our conjectures lead to results coinciding with recent work on sieved binoial polynoials, e.g., [1, 9, 12]. Naely, fro the uniodality of the coefficients in Euation (1.5), we recover the identity: the su of the coefficients of l+ in [ k 1 2 ] ( k 1 2) if k 1, 0 od for a prie is eual to 1 < k. Siilarly, we use our ore general conjecture to suggest a new identity of this type and provide an independent proof (see Proposition 8.7).

4 4 L. Lapointe and J. Morse 2. Definitions For definitions and general properties of posets see for exaple [7]. A partition λ = (λ 1,..., λ ) is a non-increasing seuence of positive integers. We denote by λ the degree λ λ of λ, and by l(λ) its length. Each partition λ has an associated Ferrers diagra with λ i lattice suares in the i th row, fro the botto to top. For exaple, λ = (4,2) =. (2.1) A partition λ is k-bounded if λ 1 k and the set of all k-bounded partitions is denoted P k. For partitions λ and µ, the weakly decreasing rearrangeent of their parts is denoted λ µ, while λ + µ is the partition obtained by suing their respective parts. Any lattice suare in the Ferrers diagra is called a cell, where the cell (i, j) is in the ith row and jth colun in the diagra. We say that µ λ when µ i λ i for all i. When µ λ, the skew shape λ/µ is identified with its diagra {(i, j): µ i < j λ i }. For exaple, λ/µ = (5, 5, 4, 1)/(4, 2) = s. (2.2) Lattice suares that do not lie inside a diagra will siply be called suares. We shall say that any s µ lies below the diagra of λ/µ. The degree of a skew-shape is the nuber of cells in its diagra. Associated to λ/µ, the hook of any (i, j) λ is defined by the cells of λ/µ that lie in the L fored with (i, j) as its corner. This definition is well-defined for all suares in λ including those below λ/µ. For exaple, the fraed cells in (2.2) denote the hook of suare s = (1, 3). We then let h s (λ/µ) denote the hook-length of any s λ, i.e., the nuber of cells in the hook of s. For exaple, h (1,3) (5, 5, 4, 1)/(4, 2) = 3 and h (3,2) (5, 5, 4, 1)/(4, 2) = 3 or cell (3, 2) has a 3-hook. We also say that the hook of a cell (or a suare) is k-bounded if it is not larger than k. A reovable corner is a cell (i, j) λ/µ such that (i + 1, j), (i, j + 1) λ/µ, and an addable corner is a suare (i, j) λ/µ such that (i 1, j), (i, j 1) λ/µ. We shall include (1, λ 1 ) λ/µ as a reovable corner, and (l(λ) + 1, 1) as an addable corner. The k-residue of any cell (or suare) (i, j) in a skew-shape λ/µ is j i od k. That is, the integer in this cell (or suare) when λ/µ is periodically labeled with 0, 1,..., k 1, where zeros fill the ain diagonal. For exaple, cell (1, 5) has 5-residue 4: Reark 2.1. No two reovable corners of any partition fitting inside the shape ( k +1 ) have the sae (k + 1)-residue for any 1 k. 3. Involution on k-bounded Partitions Usual partition conjugation defined by the colun reading of diagras does not send the set of k-bounded partitions to itself. Thus our study of P k begins with the need for a degree-preserving involution on this set that extends the notion of conjugation. Such an involution on P k was defined in [2] using a certain subset of skew-diagras. We shall follow the notation of [5], where these skew diagras are defined by:

5 Order Ideals in Sublattices of Young s Lattice 5 Definition 3.1. The k-skew diagra of a k-bounded partition λ is the skew diagra, denoted λ/ k, satisfying the conditions: (i) row i of λ/ k has length λ i, (ii) no cell in λ/ k has a hook-length exceeding k, (iii) every suare below the diagra of λ/ k has hook-length exceeding k. Exaple 3.2. Given λ = (4, 3, 2, 2, 1, 1) and k = 4, λ = = λ/ 4 =. It was shown in [5] that λ/ k is the uniue skew diagra obtained recursively by: For any λ = (λ 1,..., λ l ) P k, λ/ k can be obtained by adding to the botto of (λ 2,..., λ l )/ k, a row of λ 1 cells whose first (i.e., leftost) cell s occurs in the leftost colun where h s k. That is, row λ 1 lies as far to the left as possible without violating Condition (ii) of λ/ k, or without creating a non-skew diagra. As a atter of curiosity, with the skew diagra λ/ k = γ/ρ, it is shown in [5] that a bijection between k-bounded partitions and (k + 1)-cores arises by taking λ γ. Since the coluns of a k-skew diagra for a partition [5], and the transpose of a k-skew diagra clearly satisfies Conditions (ii) and (iii), an involution on P k arises fro the colun reading of λ/ k : Definition 3.3. For any k-bounded partition λ, the k-conjugate of λ is the partition given by the coluns of λ/ k and denoted λ ω k. Euivalently, λ ω k is the uniue k-bounded partition such that (λ/ k ) = λ ω k/ k. Corollary 3.4. For a k-bounded partition λ, we have (λ ω k) ω k = λ. Exaple 3.5. With λ as in Exaple 3.2, the coluns of λ/ 4 give λ ω 4 = (3, 2, 2, 1, 1, 1, 1, 1, 1). Given a partition λ, the nature of its k-conjugate is not revealed explicitly by Definition 3.3. However, the k-conjugate can be given explicitly in certain cases. For exaple, Reark 3.6. If h (1,1) (λ) k, all hooks of λ are k-bounded and thus λ/ k = λ. In this case, λ ω k = λ. We can also give a forula for the k-conjugate of a partition with rectangular shape. Proposition 3.7. ( n ) ω k = ( (k + 1) a, b ) where b = n od k + 1 and a = n k +1. Proof. Building the k-skew diagra recursively fro the partition ( n ) reveals that the top k + 1 rows are stacked in the shape of the rectangle = ( k +1 ). However, the (k +2)-nd row (of size ) cannot lie below any colun of without creating a

6 6 L. Lapointe and J. Morse k + 1-hook since all the coluns of have height k + 1. Thus, it lies strictly to the right of. By iteration, ( n )/ k is coprised of a seuence of block diagonal rectangles followed by a rectangular block of size ( n od k +1 ). For exaple, (3 7 ) = (3 7 )/ 4 =. (3.1) The coluns of such a skew-shape are thus as indicated. The rectangular blocks occurring in the k-skew diagra of ( n ) have the for ( k +1 ) for 1 k. We have found that such rectangles, called k-rectangles, play an iportant role in our study. For starters, we show that k-conjugation can be distributed over the union of any partition and a k-rectangle. To prove this result, we first need to find the k-conjugate of another shape: Proposition 3.8. If = (l k l+1 ) and µ (l 1) k l, then (, µ)/ k = ( + µ, µ)/µ iplying (, µ) ω k = (, µ ). (3.2) Proof. Given µ (l 1) k l, D = ( + µ, µ)/µ is a skew diagra with general shape depicted in Figure 1. If D eets Conditions (ii) and (iii) for a k-skew diagra Figure 1: ( + µ, µ)/µ, with µ depicted by the partition in horizontal stripes. then (, µ)/ k = ( + µ, µ)/µ since the rows of D are given by the partition (, µ) and thus (, µ) ω k = (, µ ) since the coluns of D are (, µ ). Any suare below D has k l + 1 cells above it and l to the right iplying it has hook k + 1 > k. On the other hand, any cell in D has hook-length strictly saller than this since the coluns and rows of D are weakly decreasing. Therefore D = (, µ)/ k. Theore 3.9. (λ ) ω k = λ ω k ω k for any k-bounded partition λ and k-rectangle. Proof. Let i be such that λ i < l and λ i 1 l, and let µ denote the non-skew partition deterined by the cells strictly above the botto row of (l, λ i,..., λ l(λ) )/ k (Figure 2). The suares in regions (a) and (3) in Figure 2 have hooks exceeding k by definition of k-skews. Now consider Figure 3 where the diagra on the left is λ/ k and in the diagra D on the right, region (b) is of shape µ. Since the rows of D are λ, if we prove that D is a k-skew diagra, then D = (λ )/ k. This will then iply (λ ) ω k = λ ω k ω k since the coluns of D are just the coluns of λ/ k and. The hooks in regions (a) and (c) of D are the sae as they are in λ/ k, and thus exceed k. Siilarly, the hooks to the right and above (a) and (c) are k-bounded in D.

7 Order Ideals in Sublattices of Young s Lattice 7 a 3 Figure 2: (l,λ i,..., λ l(λ) )/ k with µ depicted by the partition in horizontal stripes. a d c a 3 1 d b 2 c Figure 3: Coparison of λ/ k and D. The suares in region (3) lie below the k-skew diagra (l, λ i,..., )/ k and thus exceed k, as do all suares in regions 1 and d since they can only increase given that the rows of D for a partition. The cells of region (2) also exceed k since there are k l + 1 cells above the and at least l to their right. Finally, the subdiagra of D including region (b) and all cells above and to the right of this region is the diagra of ( + µ, µ)/µ. Since µ 1 < l and l(µ) k l by definition, Proposition 3.8 iplies that this subdiagra is (, µ)/ k and thus eets the conditions of a k-skew. Therefore, D is a k-skew diagra and the theore follows. Ideas to understand the nature of a k-skew diagra containing a k-rectangle that were used in Theore 3.9 ay be applied to prove the following technical proposition to be used later. Proposition For soe 1 l k, if ν is a partition containing exactly k l + 1 rows of length l, where the lowest occurs in soe row r, then there are addable corners in row r and k l r of ν/ k with the sae (k + 1)-residue. Proof. Let = (l k l+1 ) and ν = λ for soe partition λ with no parts of size l. We can construct the diagra of ν/ k as in the previous proof. That is, let i be such that λ i < l and λ i 1 > l, and let µ denote the non-skew partition deterined by the cells strictly above the botto row of (l, λ i,..., λ l(λ) )/ k. We appeal to Figure 3, where the diagra of (λ )/ k is on the right. Note that if r denotes the lowest row in ν/ k of length l, then row r 1 (if it exists) is strictly longer than row r since λ i 1 > l. Therefore an addable corner x occurs in row r with soe (k + 1)-residue j. Furtherore, since row k l r corresponds to row λ i 1 (the first row of µ) and λ i 1 < l, there is also an addable corner x in this row. Thus, it reains to show that x has (k + 1)-residue j. Let s denote the first cell in row r. If x lies in the colun of s, then the hook length of cell s is k l + l = k, and thus x also has (k + 1)-residue j. We shall now see that x does in fact lie in the colun

8 8 L. Lapointe and J. Morse containing s. If x lies in a colun to the right of s, then since x is an addable corner, the hook-length of s is larger than k (a contradiction). And if x lies in a colun to the left of s, then the suare in the row of x and the colun of x has hook-length eual to at ost k l + l = k k. 4. k-young Lattices Recall that the Young Lattice Y is the poset of all partitions ordered by inclusion of diagras, or euivalently λ µ when λ µ. Since λ µ λ µ, it is euivalent to view the Young order as λ µ when λ µ and λ µ. This interpretation for the Young order refines naturally to an order on k-bounded partitions by using the k- conjugate on P k. Definition 4.1. The order on partitions in P k is defined by the transitive closure of the relation µ λ when λ µ and λ ω k µ ω k and µ λ = 1. (4.1) Figure 4: Hasse diagra of the k-young lattice in the case k = 2. 0 We denote this poset on P k by Y k, and observe that it is a weak subposet of the Young lattice (recall this eans that if λ µ in Y k, then λ µ in Y). Furtherore, Y k reduces to the Young lattice when k since [5]: Property 4.2. λ µ reduces to λ µ when λ and µ are partitions with h (1,1) (λ) k and h (1,1) (µ) k. While this poset on k-bounded partitions originally arose in connection to a rule for ultiplying generalized Schur functions [2], it has been shown in [5] that this poset turns out to be isoorphic to the weak order on the uotient of the affine syetric group by a axial parabolic subgroup. Conseuently, Y k is a lattice [11, 13] and we thus call it the k-young lattice.

9 Order Ideals in Sublattices of Young s Lattice 9 Although the ordering is defined by the covering relation, it follows fro the definition that Property 4.3. If λ µ, then λ µ and λ ω k µ ω k. It is iportant to note that the converse of this stateent does not hold. For exaple, with k = 3, λ = (2, 2), and µ = (3, 2, 1, 1, 1, 1): λ ω k = λ and µ ω k = µ satisfy λ µ and λ ω k µ ω k, but λ µ (see Theore 4.9 and note that λ contains the 3-rectangle (2, 2) while µ does not). Since the set of µ such that µ λ and µ = λ 1 consists of all partitions obtained by reoving a corner box fro λ, the set of eleents covered by λ with respect to is a subset of these partitions. The corners that can be reoved fro λ to give partitions covered by λ are deterined as follows: Theore 4.4. [5] The order can be characterized by the covering relation λ µ µ = λ + e r, (4.2) where r is any row of µ/ k with a reovable corner whose (k+1)-residue does not occur in a higher reovable corner, or euivalently for r a row in λ/ k with an addable corner whose (k + 1)-residue does not occur in a higher addable corner. Exaple 4.5. With k = 4 and λ = (4, 2, 1, 1), λ/ 4 = , (4.3) and thus the partitions that are covered by λ are (4, 1, 1, 1), and (4, 2, 1), while those that cover it are (4, 2, 1, 1, 1) and (4, 2, 2, 1). Since the conditions of Theore 4.4 are always satisfied when choosing the reovable corner in the top row of µ we have the corollary: Corollary 4.6. If µ = (µ 1,..., µ l ) is a k-bounded partition, then λ µ for λ = (µ 1,..., µ l 1). The conditions of Theore 4.4 iply that µ = λ + e r. Therefore: Property 4.7. Any row of µ/ k containing a reovable corner whose (k + 1)-residue does not occur in a higher reovable corner, corresponds to a row of µ with a reovable corner. As discussed in the introduction, the k-rectangles play a fundaental role in the study of k-young lattices. Since k-conjugate distributes over the union of k-rectangles with a partition, and the k-young lattice relies on k-conjugates, we are able to show that the order is preserved under union with a k-rectangle. Proposition 4.8. For any k-rectangle, λ µ if and only if (λ ) (µ ). Proof. It suffices to consider the case that λ µ. Given λ µ and λ ω k µ ω k with µ λ = 1, clearly (λ ) (µ ) with µ λ = 1. Theore 3.9 then iplies that (λ ) ω k = (λ ω k ω k) (µ ω k ω k) = (µ ) ω k. That is, (λ ) (µ ).

10 10 L. Lapointe and J. Morse In fact, we have a stronger result that aounts to saying the k-rectangles play a trivial role when oving up in the k-young lattice. That is, the partitions doinating λ can be obtained by adding the parts of to the partitions that doinate λ. The (increasing) covering relations around λ and λ are isoorphic, and thus these relations are preserved under translation by a k-rectangle. Theore 4.9. For λ, µ P k and a k-rectangle, for soe k-bounded partition µ. λ µ µ = µ and λ µ, Proof. Let the k-rectangle be = (l k l+1 ). ( =) follows fro Proposition 4.8. For (= ), it suffices to consider λ µ. Theore 4.4 iplies that µ = (λ ) + e r for soe row r with an addable corner o whose (k+1)-residue does not occur in any higher addable corner of (λ )/ k. Assue by contradiction that (λ ) + e r µ for any k-bounded partition µ. The only scenario where the nuber of rows of length l is reduced by adding a box is if λ has exactly k l + 1 rows of length l and row r is the lowest row of length l. Thus, by Proposition 3.10, there is an addable corner in row k l r of (λ )/ k with the sae (k + 1)-residue as o. However, row k l r is higher than row r and by contradiction, µ = µ for soe µ. Finally, given λ µ, the previous proposition iplies λ µ. Reark Consider λ, µ P k and a k-rectangle. Notice that in general, λ µ λ = λ and λ µ. For exaple, with k = 3: the 3-rectangle (2, 1) (2, 2) while (2, 1) (2, 2). However, µ and λ µ λ = λ and λ µ follows fro Proposition 4.8. In this case, what occurs above and below µ is replicated at µ. Interpreting the poset as a cone in a tiling of k-space by perutahedrons [11], this iplies that a vertex µ lying at least a distance fro the boundary of the cone can not be distinguished fro µ, and thus the k-rectangles are the vectors of translation invariance in the tiling. 5. Principal Order Ideal Let Y λ denote the principal order ideal generated by λ in the Young lattice. When λ is a rectangle, the order ideal is denoted L(, n) and is the induced poset of partitions with at ost n parts and largest part at ost. This order ideal is a graded, self-dual, and distributive lattice. Further, the rank-generating function of L(, n), the Gausssian polynoial, is known to be uniodal. The next several sections concern the study of properties for order ideals in the k-young lattice that are analogous to those held by L(, n). Let Y k λ denote the principal order ideal generated by λ in the poset Y k. That is, Y k λ = {µ: µ λ}. As k, the poset Y k λ reduces to Y λ. More precisely,

11 Order Ideals in Sublattices of Young s Lattice 11 Property 5.1. If λ is a partition with h (1,1) (λ) k, then Y k λ = Y λ. Proof. Since µ λ iplies that µ λ, we have that h (1,1) (µ) h (1,1) (λ) k. Thus, for all µ, ν Y k λ we have µ ν µ ν by Property 4.2, that is, Y k λ = Y λ. Proposition 5.2. Yλ k is graded of rank λ. Proof. If λ (1), λ (2),...λ (n) is a saturated chain in Yλ k then λ(i) = λ (i+1) 1 fro the definition of the order. Corollaries 4.6 iplies that a axial chain in Yλ k ust begin with the epty partition. Therefore, since by definition of Yλ k all axial chains start with λ, we have our clai. We are interested in proving properties of Yλ k when λ is a rectangular partition, denoted by: Definition 5.3. The principal order ideal of Y k generated by the partition n will be denoted L k (, n) = {µ: µ ( n )}. (5.1) Note that k since Y k contains only eleents of P k. Further, Reark 5.4. Fro Property 5.1, L k (, n) = L(, n), for n k + 1. Therefore, all cases are covered when considering n k + 1. That is, all L k (, n) distinct fro L(, n), plus the non distinct case L n+ 1 (, n) = L(, n). We shall prove that L k (, n) is a graded, self-dual, and distributive lattice. Further, we shall conjecture that its rank-generating function is uniodal in certain cases. To start, following fro Proposition 5.2, Corollary 5.5. L k (, n) is graded of rank n. It will develop that for each k, the principal order ideal generated by ( n ) in Y k is isoorphic to an induced subposet of the principal order ideal generated by ( n ) in the Young Lattice. Fro this, we can then derive a nuber of properties for the posets L k (, n). To this end, we first explicitly deterine the vertices of the order ideal. Theore 5.6. The set of partitions in L k (, n) are those that fit inside an n rectangle and have no ore than k + 1 rows shorter than. Proof. We first show that any λ L k (, n) can have at ost k + 1 rows of length shorter than. Suppose the contrary, and note that the top k + 2 rows of λ/ k for a partition since the rows are all shorter than and thus no hook exceeds k. Therefore, the first colun of λ/ k has height at least k + 2 and λ ω k has a row of length at least k + 2 by definition of k-conjugate. However, the rows of ( n ) ω k do not exceed k +1 by Proposition 3.7 and thus λ ω k ( n ) ω k. Therefore λ L k (, n) by Property 4.3. On the other hand, to prove that any λ = ( a, µ) n with µ ( k +1 ) lies in L k (, n), it suffices to prove λ L k (, l(λ)) since l(λ) n i.e., using Corollary 4.6 ties, we obtain n 1 n, and by iteration l(λ) n. To this end, note that the top l(µ) k + 1 rows of ( l(λ) )/ k fit inside the shape ( k +1 ) iplying every diagonal has a distinct (k + 1)-residue by Reark 2.1. Therefore, Theore 4.4 iplies reovable corners can be successively reoved fro the top l(µ) rows in ( l(λ) ) to obtain partitions λ (1),..., λ (i) where λ λ (1) λ (i) ( l(λ) ).

12 12 L. Lapointe and J. Morse The theore reveals that when n k + 1, the eleents of L k (, n) are of the for ( a, µ) for µ ( k +1 ) and a n (k + 1). By Reark 5.4, we have thus identified all the vertices. Corollary 5.7. For n k + 1 and k, the set of partitions in L k (, n) is the disjoint union { µ ( k +1 ) } n (k +1) i=1 { ( i, µ 1 + 1,..., µ k ) : µ ( 1) k +1}. It also follows fro Theore 5.6 that the vertices of L (, n) are siply the partitions with at ost one row saller than. Corollary 5.8. The set of partitions in L (, n) is L (, n) = { ( j, i) : 0 i and 0 j n 1 }. (5.2) 6. Further Properties of k-young Lattice Ideals Euipped with a siple characterization of the vertex set of L k (, n), we can now investigate a connection between the k-young lattice and the Young lattice. As it turns out, the principal order ideal generated by ( n ) in Y k is isoorphic to the induced subposet of L(, n) containing only the subset of partitions with no ore than k +1 rows saller than. Conseuently, it is easy to grasp which eleents are covered by λ in L k (, n) and to deduce that the posets are self-dual and distributive. Proposition 6.1. Let λ, µ L k (, n). Then µ λ if and only if µ < λ. Proof. First consider λ L k (, n) where λ ( k +1 ). Since h (1,1) (λ) k, µ λ reduces to µ λ by Property 4.2. Any other eleent of L k (, n) has the for λ = ( b, ν) for soe ν P with l(ν) = k + 1 by Corollary 5.7. Given λ of this for, since µ λ and µ < λ both reuire that µ = λ e r where r is a row of λ with a reovable corner, we need only consider b r b + l(ν). Further, (λ e r ) L k (, n) if r = b (the partition would have ore than k + 1 rows shorter than ). Thus it suffices to show that for b < r b + l(ν), there is a reovable corner in row r of λ if and only if (λ e r ) λ euivalently by Theore 4.4 if and only if there is a reovable corner in row r of λ/ k whose (k + 1)-residue does not occur in any higher reovable corner. When row r of λ/ k has a reovable corner that is the highest of a given (k + 1)- residue, there is a reovable corner in row r of λ by Property 4.7. On the other hand, if there is a reovable corner in row b < r b + l(ν) of λ then there is also a reovable corner in this row of λ/ k since the top l(ν) rows of ( b, ν)/ k coincide with the diagra of ν given that l(ν) = k + 1. Furtherore, ν ( k +1 ) also iplies that there cannot be another reovable corner above the one in row r of the sae (k + 1)-residue because the diagonals in ( k +1 ) all have distinct (k + 1)-residue.

13 Order Ideals in Sublattices of Young s Lattice 13 Theore 6.2. L k (, n) is isoorphic to the induced subposet of L(, n) with vertices restricted to the partitions in L k (, n). Euivalently, for λ, µ L k (, n), µ λ µ λ. (6.1) Proof. Let λ, µ L k (, n). Fro Property 4.3, µ λ = µ λ. It thus reains to show that there exists a chain fro µ to λ in L k (, n) when µ λ. Euivalently by the previous proposition, it suffices to show that we can reach µ by successively adding boxes to λ in such a way that no interediate step gives a partition with ore than k + 1 rows of length less than. This is achieved as follows: given µ λ in L k (, n), consider the chain of partitions µ = µ 0 µ 1 µ j = λ where µ i+1 is obtained by adding one box to the first row in µ i that is strictly less than the corresponding row in λ. Since the chain starts fro µ = ( a, ν) where l(ν) k + 1, the nuber of rows of length less than does not exceed k + 1 in any µ i by construction. We can now derive a nuber of properties for the order ideals L k (, n) based on the identification with induced subposets of L(, n) under inclusion of diagras. First, given the explicit description Euation (5.2) for the vertices in L (, n) we have Proposition 6.3. The order ideal L (, n) is isoorphic to the saturated chain of partitions: /0 (1) () (, 1) ( j, i) ( n 1, 1) ( n ). Proposition 6.4. For k, L k (, n) is an induced subposet of L k+1 (, n). Proof. Fro Theore 5.6, the eleents of L k (, n) (or L k+1 (, n)) are partitions contained in ( n ) with at ost k + 1 (resp. k + 2) rows that are saller than. Therefore, by Theore 6.2, it suffices to note that under inclusion of diagras, the poset of partitions that fit inside an n rectangle with no ore than k + 1 parts saller than is an induced subposet of the poset of partitions that fit inside an n rectangle with no ore than k + 2 parts saller than. Proposition 6.5. Let λ, µ L k (, n). (i) λ µ is the partition deterined by the intersection of the cells in the diagras in λ and µ. (ii) λ µ is the partition whose diagra is deterined by the union of the cells in λ and µ. Proof. Since the eet and join of eleents in L(, n) is given by the intersection and union of diagras respectively, and L k (, n) is isoorphic to an induced subposet of L(, n) by Theore 6.2, it suffices to show that L k (, n) is closed under the intersection and the union of diagras. Euivalently, by Theore 5.6, we ust prove that the intersection and union of such partitions do not have ore than k + 1 rows with length less than. Let λ = ( a, λ) with l( λ) k + 1 and µ = ( b, µ) with l( µ) k + 1 and assue a b without loss of generality. As such, the diagra of λ intersected with µ has at least b rows of length, and at ost l( µ) rows with length less than. Therefore, there are no ore than k + 1 rows of length saller than. Siilarly, the union has at least a rows of length, and at ost ax { l( λ), l( µ) (a b) } rows less than. Again, no ore than k + 1 rows of length saller than.

14 14 L. Lapointe and J. Morse Now given that the eet and join of L k (, n) coincide with those of L(, n), therefore, since L k (, n) is an induced subposet of the lattice L(, n), we have 1 Corollary 6.6. For each k, L k (, n) is a lattice. Furtherore, since L(, n) is a distributive lattice, the relations λ (µ ν) = (λ µ) (λ ν), λ (µ ν) = (λ µ) (λ ν), (6.2) ust hold. Therefore, these relations hold in the induced subposets L k (, n) and we find Corollary 6.7. For each k, L k (, n) is distributive. In addition to having that each of the L k (, n) are distributive lattices, we can also prove that they are syetric. Theore 6.8. For any k, L k (, n) is self-dual. Proof. Let L k (, n) denote the dual of L k (, n) and consider the apping φ(λ) = λ c where λ c is the partition deterined by rotating the copleent of λ in ( n ) by 180. Since λ L k (, n) iplies that λ = ( a, µ) for soe a n (k + 1) and µ ( k +1 ) by Corollary 5.7, λ c = ( n (k +1) a, µ c ) for soe µ c ( k +1 ) satisfying λ + λ c = n. Therefore φ: L k (, n) L k (, n) and it suffices to show that φ is an order-preserving bijection. Euivalently, that µ λ in L k (, n) φ(λ) φ(µ) in L k (, n). By Theore 6.2 and the definition of L k (, n), this is euivalent to µ λ λ c µ c for eleents λ, µ L k (, n), which is true. Corollary 6.9. L k (, n) is rank-syetric. 7. Rank-Generating Function The explicit description of the partitions in the posets L k (, n) can also be used to deterine the rank-generating functions. Recall that the nuber of eleents of rank i in L(, n) is the coefficient of i in the Gaussian polynoial: n p i (, n) i = [ ] + n. (7.1) Siilarly, we can deterine the rank-generating functions for L k (, n). Theore 7.1. For n k + 1 and k, the nuber of eleents of degree i in L k (, n) is the coefficient of i in n p k i (, n)i : = [ ] k k+1 1 (n k+ 1) 1 1 Note that the corollary also follows iediately fro the fact that Y k is a lattice. [ ] k. (7.2) 1

15 Order Ideals in Sublattices of Young s Lattice 15 Proof. Recall that Corollary 5.7 provides an interpretation for the vertices of L k (, n) as a disjoint union of sets whose eleents can be understood in ters of certain L(a, b): { µ ( k +1 ) } n (k +1) i=1 { ( i, µ 1 + 1,..., µ k ) : µ ( 1) k +1}. Identity (7.1) then gives that the nuber of eleents of rank i in L k (, n) is the coefficient of i in [ ] [ ] k + 1 k + (i)+k +1. (7.3) 1 n (k +1) i=1 Letting 1 in Euation (7.3) then gives the nuber of vertices: Corollary 7.2. For n k + 1 and k, the nuber of vertices in L k (, n) is n ( ) ( ) p k k + 1 k i (, n) = + (n k + 1). (7.4) 1 In the next section, we will see that the study of the order ideals L k (, n) is siplified by a close exaination of certain subsets of the vertices that are deterined by k. Definition 7.3. For n k +1 and k >, let Γ k (, n) denote the set of all partitions in L k (, n) that are not in L k 1 (, n). We conclude the section with a discussion of the sets Γ k (, n), starting with an explicit description of the eleents. Proposition 7.4. The eleents of Γ k (, n) are partitions that fit inside an n rectangle and have exactly k + 1 rows with length saller than. Euivalently, Γ k (, n) { = ( a, µ 1 + 1,..., µ k ): 0 a n (k + 1), µ () k +1} (7.5) Proof. Theore 5.7 indicates that the vertices of L k (, n) are partitions with no ore than k + 1 rows of length saller than while those of L k 1 (, n) are partitions with no ore than k rows of length saller than. The result thus follows. Since the vertices of L (, n) are given in Euation (5.2) to be the partitions with no ore than one row of length saller than, the set of vertices for L k (, n) can be obtained by adding the eleents of Γ i (, n) for i = +1,..., k to this set of partitions. That is, Proposition 7.5. For n k + 1 and k >, the vertices of L k (, n) are the disjoint unions: L k (, n) = L (, n) k j=+1 Γ j (, n). (7.6)

16 16 L. Lapointe and J. Morse Given this decoposition for the set of vertices in L k (, n), we note that there are no edges between partitions in Γ j (, n) and partitions in Γ j i (, n) for any i > 1 since adding or deleting a box to a partition with exactly a rows saller than length never gives rise to a partition with ore than a + 1 rows saller than. Now, L +1 (, n) can be constructed by connecting eleents of Γ +1 (, n) to the appropriate vertices in the saturated chain L (, n) of Proposition 6.3. L +2 (, n) can then be constructed by connecting only eleents of Γ +1 (, n) to the appropriate eleents of Γ +2 (, n). In this anner, the poset L k (, n) can be constructed fro the saturated chain L (, n). In particular, L(, n) = L +n 1 (, n) can be obtained using this process (see Figure 5 for an exaple). In suary, Reark 7.6. The poset L k (, n) can be obtained fro L k 1 (, n) by adding edges fro λ Γ k 1 (, n) to all partitions in Γ k (, n) that contain or are contained in λ. Figure 5: Decoposing L 5 (3, 3) into L 3 (3, 3) Γ 4 (3, 3) Γ 5 (3, 3).

17 Order Ideals in Sublattices of Young s Lattice 17 We now obtain the rank-generating function for Γ k (, n) fro Proposition 7.4 by using the Gaussian polynoial for partitions inside a (k + 1) () rectangle. Proposition 7.7. For n k +1 and k >, the rank-generating function for Γ k (, n) is u k i (, n) i := λ = k +1 (1 [ ] (n k+) ) k 1 (1. (7.7) ) i 0 λ Γ k (,n) In this euation, u k i (, n) is defined to be the nuber of eleents of degree i in Γk (, n). It turns out that the seuence of coefficients in this expression is rank-syetric. Proposition 7.8. u k i (, n) = uk n n i (, n) for all i = 0,..., 2. That is, the vector of coefficients u k = (u k 0 (, n),..., uk n (, n)) is syetric about the iddle (i.e., n/2). Proof. By Euation (7.7), u k i (, n) is the coefficient of i in a product of three ranksyetric polynoials: k +1, (1 (n k+) ) (1 ), and [ k 1] 2. Therefore, since the ter of lowest degree in the expansion of these three polynoials has degree k + 1 and the highest degree ter has degree k +1+(n k + 1)+( 2)(k +1) = n k + 1, the polynoial ust be rank-syetric about ( k (n k + 1) ) /2 = n/2. 8. Uniodality and Sieved Sus Recall that a poset P of rank d is uniodal if p 0 p 1 p i p i+1 p d for soe 0 i d, where p i is the nuber of eleents with rank i in P. We shall say the rank-generating function for P is uniodal when the vector (p 0,..., p d ), with p i the coefficient of i, fors a uniodal seuence. For exaple, the Gaussian polynoial (7.1) is uniodal because it is known [6,10] that the coefficients (p 0,..., p n ) for a uniodal seuence. Euivalently, L(, n) is a uniodal poset one of the deeper properties of this order ideal. In this section, we address the uestion of uniodality for the posets L k (, n), illustrating the role of k in our study. As such, by Euation (7.2), the order ideal L k (, n) will be uniodal when n p k i (, n)i = [ ] k k+1 1 (n+ k 1) 1 [ ] k 1 (8.1) is uniodal. Our work in the preceding section pays off here by enabling us to rewrite this expression as a su of rank-syetric coponents. Proposition 8.1. The nuber of eleents in L k (, n) is given by the coefficient of i in ( ) n p k i (, n) i n k = 1 + u r i (, n) i, (8.2) r=+1 where u r i (, n) is defined in Euation (7.7).

18 18 L. Lapointe and J. Morse Proof. Proposition 7.5 decoposes the set of vertices of L k (, n) into L (, n) k i=+1 Γ i (, n), iplying that the coefficients p k i (, n) occur in the expression: n p k i (, n)i = λ L (,n) λ + k r=+1 λ Γ r (,n) λ. (8.3) The result then follows since L (, n) is just a saturated chain by Proposition 6.3, and the eleents in Γ r (, n) are given in Proposition 7.7 by u r i (, n). Since u r i (, n) = ur n i (, n) by Proposition 7.8, (8.2) reveals p k (, n) = (p k 0 (, n),..., p k n (, n)) is a su of seuences syetric about the iddle. Thus, the uestion of uniodality of L k (, n) reduces to a study of the seuences u r (, n). We have experientally discovered that certain sus of these seuences are uniodal under conditions on expressed in Conjecture 8.2, and ore generally in Conjecture 8.5. This given, we can deduce that p k (, n) is uniodal in these cases and can be written as a su of rank-uniodal seuences (see Reark 8.6). To this end, we start with the special case when is prie. Conjecture 8.2. If n k +1 and k 1, 0 od for a prie nuber < k, then n [ ] u k i (, n)i k +1 1 (n k+) k 1 = 1, (8.4) is uniodal. Further, when n > k + 1 and k = 1 od, n ( ) u k i (, n) + uk+1 i (, n) i k +1 1 (n k+) = 1 is uniodal. [ ] k 1 + k +2 1 (n k+ 1) 1 [ ] k, (8.5) There are two conseuences of this conjecture; the first relating to the uniodality of L k (, n). Conseuence of Conjecture 8.2. If n > k + 1 and k 1 od for a prie < k, then the order ideal L k (, n) is uniodal. Euivalently, n [ ] [ ] k + 1 p k i (, n)i k+1 1 (n+ k 1) k = + 1 (8.6) 1 is uniodal under these conditions. Proof following fro Conjecture 8.2. By Euation (8.2), it suffices to show that k j=+1 u j (, n) is uniodal. For all < j < n + 1, the seuences u j (, n) are ranksyetric about n/2 by Proposition 7.8. Therefore, if u j (, n) is uniodal for j 1, 0 od and u j (, n) + u j+1 (, n) is uniodal when j = 1 od, then k j=+1 u j (, n) is uniodal unless k = 1 od. That is, except in the case that u j+1 (, n) cannot be added to restore uniodality.

19 Order Ideals in Sublattices of Young s Lattice 19 Interestingly, a second conseuence of Conjecture 8.2 relates to sieved -binoial coefficients (see for exaple, [1, 9, 12]). In this result, the su of the coefficients of l+ in an expression refers to the su of coefficients of i for every i = l od. Proposition 8.3. If k 1, 0 od for a prie < k, then for any 0 l 1, the su of the coefficients of l+ in [ k 1 2 ] is ( k 1 2) /. Euivalently, p l+ j (, k + 1) = 1 ( ) k 1, for fixed l = 0,..., 1. (8.7) j 0 We shall deonstrate how this proposition follows fro Conjecture 8.2 to give evidence supporting the conditions under which we believe uniodality to hold. However, the result is iplied fro the ain result in [12] (and ore recently fro [9]) as follows: Proof. Condition 3 of the ain result in [12] says that the su of the coefficients of l+ in [ n t ] is ( n t) / for all l iff for all d > 1 that divide, we have t od d > n od d. Since only d = divides when is prie, letting n = k 1 and t =, we note that od > k 1 od is euivalent to the condition k 1, 0 od. Proof following fro Conjecture 8.2. For k 1, 0 od for prie, Conjecture 8.2 iplies 1 (n k+) [ ] k 1 1, is uniodal. Letting n in this expression then produces an increasing (uniodal and syetric about infinity) seuence (v k 0 (), vk 1 (),...), where [ ] k 1 v k i () i = ( ) i. (8.8) Using the definition of p i (, n) in Euation (7.1), the coefficient of i in this expression satisfy v k i () = p i( 2, k +1)+ p i ( 2, k +1)+ p i 2 ( 2, k +1)+. This seuence thus eets the conditions in the following proposition with a = p( 2, k + 1), where the clai follows fro Euation (8.9) since p i (, k + 1) = i ( ) k 1. Proposition 8.4. Let v i = t 0 a i t for soe a = (a 0,..., a D ) and a j = 0 when j < 0 or j > D. If the infinite seuence v = (v 0, v 1,..., ) is weakly increasing, then (i) v i = v D +1 for i D + 1. (ii) For any l = 0,..., 1,

20 20 L. Lapointe and J. Morse j 0a l+ j = 1 D j=0 a j. (8.9) Proof. (i) Since i+ > D when i D +1, we have a i+ = 0. The definition of v i+ iplies v i+ = a i+ + v i and thus v i+ = v i when i D + 1. To show v i = v D +1 for all i D + 1, note that v D +1 v D +2 v D+1 since v is increasing. However, v D +1 = v D+1 by the preceding arguent, iplying v D +1 = v D +2 = v D+1. By iteration, v i = v D +1 for all i D + 1. (ii) Since a i = 0 for i < 0, we have for all 1 l that v D l+1 = a D l+1 + a D l a D l+1 (D l+1)/. (8.10) That is, v D l+1 = n a n for n = D l + 1 od. The su v D v D 1 + v D is thus the su of all entries of a. However, (i) iplies that v D +1 = = v D 1 = v D and therefore, v D l+1 = D j=0 a j, for l = 1,...,. Fro Euation (8.10), v D l+1 is the su of entries in a indexed by D l + 1 odulo, thus iplying the clai since as l runs over 1 to, D l + 1 runs over all possible values odulo. In suary, when is prie, proving uniodality of the posets L k (, n) reduces to proving the uniodality of u r (, n) + u r+1 (, n). As such, to extend this idea for not prie, we study the uniodality of ore general sus of u r (, n). Conjecture 8.5. Consider integers a and b such that a < b n + 1. If a, b 1 od p for every prie divisor p of, then n is uniodal. ( b u j i ) (, n) i = j=a+1 b j=a+1 j +1 1 (n j+) 1 [ ] j 1, (8.11) We recover Conjecture 8.2 by taking prie and letting a = k 1 and b = k to get the uniodality of u k (, n). Further, a = k 1 and b = k +1 iplies the uniodality of u k (, n) + u k+1 (, n). As with the case of prie, we can extract two conseuences of Conjecture 8.5. Conseuence of Conjecture 8.5. Let k >. If k 1 od p for every prie divisor p of, then L k (, n) is uniodal for all n > k + 1. Euivalently, [ ] k is uniodal under these conditions. k+1 1 (n+ k 1) 1 [ ] k, (8.12) 1 Proof following fro Conjecture 8.5. Given k 1 od p for all prie divisors p of and noting that 1 od p, we eet the conditions of Conjecture 8.5 with a = and b = k. Therefore, p k (, n) is uniodal using the decoposition given in Proposition 8.1.

21 Order Ideals in Sublattices of Young s Lattice 21 Reark 8.6. Following trivially fro Proposition 8.1, we can obtain an alternative decoposition for p k (, n) in ters of sus of u r (, n): for nonnegative integers k 1 < k 2 < < k l = k, when each k j 1 od p for all prie divisors p of k, n p k i (, n)i ( = n 1 + k 1 u r i r=+1 (, n) + k 2 r=k 1 +1 u r k i (, n) + + l u r i ) (, n) i. (8.13) r=k l 1 +1 Thus, by Conjecture 8.5, the order ideals L k (, n) can be decoposed in ters of l + 1 uniodal pieces rank-syetric about the sae point n/2. Our second conseuence is a ore general result on sieved -binoial coefficients. We state this conseuence as a proposition since we will provide a proof that is independent of our conjecture. Proposition 8.7. If a < b are non-negative integers where a, b 1 od p for every prie divisor p of, then the su of the coefficients of l+ in b j=a+1 j (a+1) [ j 1 ] ( j 1 2 2), for any 0 l 1. is 1 b j=a+1 Note that even when is prie, this is ore general than Proposition 8.3. For exaple, the case that k = 1 od is included when a = k 1 and b = k +1. Before proving this result, we shall show how it follows fro Conjecture 8.5 to support the validity of our conjecture. Proof following fro Conjecture 8.5. Letting n in Euation (8.11), (and ultiplying by a 2 for convenience), the coefficients (v a,b 0 (), va,b 1 (),... ) defined by v a,b i () i = ( ) i 0 b j=a+1 for an increasing seuence given Conjecture 8.5. Thus, defining we have v a,b i [ ] j 1 j (a+1), (8.14) p a,b i () i b [ ] j 1 = j (a+1), (8.15) i 0 j=a+1 () = p a,b () + p a,b () + pa,b i i i 2 The proposition thus follows fro Proposition 8.4 with a = (p a,b 0 since i p a,b i () = b ( j 1 j=a+1 2). Proof. Fro {, if r, ω r = ω: ω =1 0, otherwise, () +. (8.16) (), pa,b 1 (),...), (8.17)

22 22 L. Lapointe and J. Morse we have that the su of coefficients of l+ in any polynoial P() is 1 ω l P(ω). (8.18) ω: ω =1 Thus, to prove the su of the coefficients of l+ in b j=a+1 j (a+1)[ ] j 1 2 is b j=a+1 ( j 1 2) /, it suffices to prove 1 b j=a+1 [ j 1 ω (l j+a+1) ω: ω =1 ] ω = 1 b j=a+1 ( ) j 1. (8.19) Or euivalently, since the right hand side euals the ω = 1 ter in the left hand side, b [ ] 1 j 1 ω (l j+a+1) = 0. (8.20) j=a+1 ω =1 ω 1 To this end, we shall deonstrate that for all ω such that ω = 1 and ω 1, b [ ] j 1 ω j = 0. (8.21) j=a+1 ω by proving this identity holds when ω is a priitive d th root of unity, for all d not eual to 1. If j 1, 0 od d, then [ j 1 2 ]ω = 0 since the nuerator of [ ] j 1 2 has one ore ω zero than its denoinator, given that i od d = 0 for soe j d + 2 i j 1. On the other hand, when j = 1 od d (see [9, Lea 1(3)]) [ ] ( ) j 1 n 1 =, (8.22) /d 1 for n = ( j + 1)/d. Thus, using [ ] j = 1 ω j [ ] j 1 ω 1 ω j +2, ω we also have [ ] j ω ω ω = 1 ( ) ω 1 n 1. (8.23) 1 ω /d 1 However, the conditions a, b 1 od p for every prie divisor p of iply that a, b 1 od d for any d. Therefore, the cases a+1 = 0 od d, and b = 1 od d cannot be liits in the su (8.21), iplying that if a ter of the for j = 1 od d occurs in the su then so does j + 1 = 0 od d. Therefore, since these pairs of ters satisfy [ ] [ ] ( ) j 1 ω j + ω j+1 j n 1 = ω ( ) ω 1 n 1 = 0, /d 1 1 ω /d 1 ω the proposition holds. ω Acknowledgent. We thank Dennis Stanton for his uestions that initiated part of this work and for several very helpful references.

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