P -PARTITIONS AND RELATED POLYNOMIALS

Transcription

1 P -PARTITIONS AND RELATED POLYNOMIALS 1. P -artitions A labelled oset is a oset P whose ground set is a subset of the integers, see Fig. 1. We will denote by < the usual order on the integers and < P the artial order defined by P. A function σ : P Z + {1, 2, 3,...} is a P -artition if (1) σ is order reserving, i.e., σ(x) σ(y) whenever x P y; (2) If x < P y in P and x > y, then σ(x) < σ(y). Let A(P ) be the set of all P -artitions and if n Z + let A n (P ) {σ A(P ) : σ(x) n for all x P }. The order olynomial of P is defined for ositive integers n as Ω(P, n) A n (P ). A linear extension of P is a total order, L, on the same ground set as P satisfying x < P y x < L y. Hence, if L is a linear extension of P we may list the elements of P as x 1 < L x 2 < L < L x, where is the number of elements in P. The Jordan Hölder set of P is the following set of ermutations of P coming from linear extensions of P : L(P ) {x 1 x 2 x n : x 1 < L < L x is a linear extension of P }. Proosition 1.1. Let P be a finite labelled oset. Then A(P ) is the disjoint union A(P ) L A(L), (1) where the union is over all linear extensions of P. Proof. If L and L are two different linear extensions of P then there are x, y P for which x comes before y in L and y comes before x in L. Suose x > y as integers. Then σ(x) < σ(y) for all σ A(L) and σ (x) σ (y) for all σ A(L ). Figure 1. A labelled oset. 1

2 2 P -PARTITIONS Clearly A(P ) L A(L). I remains to rove the converse inclusion. If P is a total order then there is nothing to rove. Suose that x, y P are unrelated and x > y. Let P xy be the oset with the same ground set as P defined by z Pxy w if and only if z P w, or z P x and y P w. Clearly A(P ) A(P xy ) A(P yx ). If σ A(P ), then either σ(x) < σ(y) or σ(x) σ(y). In the first case σ A(P xy ) and in the other σ A(P xy ), and hence the union above is disjoint. Thus A(P ) A(P xy ) A(P yx ). If P xy or P yx is not a total order we may continue this rocess until we get a disjoint union over linear extensions. The following corollary roves that Ω(P, n) is indeed a olynomial, and may thus be defined for negative integers or for any comlex number n. Corollary 1.2. Let P be a finite oset of cardinality. Then Ω(P, n) n + des(π) 1. In articular Ω(P, n) is a olynomial in n of degree, with leading coefficient equal to e(p )/!, where e(p ) is the number of linear extensions of P. Proof. By Proosition 1.1 Ω(P, n) Ω(P π, n), where P π is the total order π 1 < L < L π. sequences Now, Ω(P π, n) is the number of 1 σ 1 σ n, where σ j < σ j+1 whenever π j > π j+1. Let τ j σ j + {i : i < j and π i < π i+1 }. Then we see that Ω(P π, n) is the number of sequences 1 τ 1 < τ 2 < < τ n + 1 des(π), which roves the corollary. Suose that P is a labelled oset. Let P be the oset on { x : x P }, with artial order defined by x P y if and only if x P y. Theorem 1.3 (Recirocity). Let P be a labelled oset of cardinality and let n be a ositive integer. Then Ω(P, n) ( 1) Ω( P, n). Proof. For π L(P ) let π be defined π ( π 1 )( π 2 ) ( π d ). Then π L(P ) if and only if π L( P ). Since des(π) 1 des( π) we have by Corollary

3 P -PARTITIONS Ω( P, n) ( 1) Ω(P, n) The P -Eulerian olynomial is defined by W (P, t) n + des( π) 1 n + des(π) n + des(π) 1 ( 1) t des(π)+1. Theorem 1.4. Let P be a finite labelled oset of cardinality. Then Ω(P, n)t n W (P, t) (1 t) +1. Proof. In view of Corollary 1.2 it suffices to rove that n + k t n t k (1 t) +1, for all k 1. This is a consequence of the binomial theorem for negative exonents. In terms of P -Eulerian olynomials, Theorem 1.3 translates as W ( P, t) t W (P, 1/t) (2) Corollary 1.5. Let d be a ositive integer. Then π S n d t n d t des(π)+1 (1 t) d+1. Proof. Let A d be the anti chain on {1,..., d}. Then Ω(A d, n) n d and L(A d ) S d. Aly Theorem Sign-graded osets Let P be a finite labelled oset, and let E(P ) be the set of covering relations (Hasse-diagram) of P. Define a function ɛ P : E(P ) { 1, 1} as follows. { 1 if x < y, ɛ P (x, y) 1 if x > y. Remark 1. Note that a function σ : P Z + is a P -artition if and only if for all (x, y) E(P ). σ(y) σ(x) ɛ P (x, y)/2

4 4 P -PARTITIONS Figure 2. A sign graded oset oset and its rank function. The labelled oset P is sign-graded of rank r(p ) if m ɛ P (x k 1, x k ) r(p ) k1 for all maximal chains x 0 < x 1 < < x m in P. If P is sign graded we define a rank function ρ P : P Z by l ρ P (x) ɛ P (x k 1, x k ), k1 where x 0 < x 1 < < x l x is any unrefinable chain from a bottom element to x. Note that ρ P (y) ρ P (x) ɛ P (x, y) if y covers x. Note also that ρ P (x) l mod 2. (3) Remark 2. A labelled oset, P, is called natural if x P y imlies x y. Hence for natural labelled osets ɛ P (x, y) 1 for all (x, y). Thus a natural labelled oset is sign graded if and only it is graded in the usual sense. Theorem 2.1. Let P and Q be sign graded osets of rank r(p ) and r(q), resectively. If P and Q are isomorhic as osets, then ( ) r(p ) r(q) Ω(P, n) Ω Q, n +. 2 Proof. Suose that x x defines an isomorhism between P and Q. By (3) it follows that ρ P (x) ρ Q (x ) mod 2 for all x P. Define a function ξ : A(P ) Z Q by ξ(σ)(x ) σ(x) + (ρ P (x) ρ Q (x ))/2. We will show that ξ is a bijection between A n (P ) and A n+µ (Q), where µ (r(p ) r(q))/2. Suose that y covers x in Q. Then by Remark 1 ξ(σ)(y ) ξ(σ)(x ) σ(y) σ(x) + (ρ P (y) ρ Q (y ))/2 (ρ P (x) ρ Q (x ))/2 σ(y) σ(x) + (ρ P (y) ρ P (x))/2 (ρ Q (y ) ρ Q (x ))/2 σ(y) σ(x) + ɛ P (x, y)/2 ɛ Q (x, y )/2 ɛ Q (x, y )/2. In articular, ξ(σ) is order reserving. Hence ξ(σ) A(Q) rovided that ξ(σ)(x ) 1 for all minimal elements x Q. However, ξ(σ)(x ) σ(x) 1 if x is minimal.

5 P -PARTITIONS 5 Moreover, if x is maximal then ξ(σ)(x ) σ(x) + µ. Thus ξ : A n (P ) A n+µ (Q). Clearly ξ is invertible with inverse given by ξ 1 (σ)(x) σ(x ) + (ρ Q (x ) ρ P (x))/2. Combining Theorems 1.3 and 2.1 we obtain: Corollary 2.2. Let P be a sign graded osets of rank r. Then Ω(P, n) ( 1) Ω(P, n r). Proosition 2.3. Let P and Q be sign graded osets of rank r(p ) and r(q), resectively. If P and Q are isomorhic as osets, then W (Q, t) t (r(p ) r(q))/2 W (P, t). Proof. By the roof of Theorem 1.4 we see that if Ω(P, n) n + k w k (P ), k 1 then W (P, t) k 0 w k (P )t k. Let µ (r(p ) r(q))/2. By Theorem 2.1 Ω(P, n) Ω(Q, n + µ) n + µ + k w k (Q) k n + j w j+µ (Q). j Hence w j (P ) w j+µ (Q) for all j and the roosition follows. Combining (2) and Corollary 2.3 we see that the P -Eulerian olynomial of a sign-graded oset is alindromic: t r(p ) W (P, 1/t) W (P, t). Deartment of Mathematics, Royal Institute of Technology, SE Stockholm, Sweden address: branden@kth.se