A Rook Theory Model for the Generalized p, q-stirling Numbers of the First and Second Kind

Transcription

1 Formal Power Series Algebraic Combinatorics Séries Formelles et Combinatoire Algébriue San Diego, California 2006 A Rook Theory Model for the Generalized p, -Stirling Numbers of the First Second Kind Karen Sue Briggs Abstract. In (EJC 11 (2004), #R84), Remmel Wachs presented two natural ways to define p, - analogues of the generalized Stirling numbers of the first second kind, S 1 (α, β, r) S 2 (α, β, r) as introduced by Hsu Shiue (Adv. App. Math 20 (1998), ). In this paper, we present a rook theoretic model for each type of p, -analogue based on a pair of boards parametrized by the nonnegative integers α, β, r, so that rooks attack cells on its own board as well as on its companion board. For each model, we provide an analogue of Goldman, Joichi White s product formula (Proc. Amer. Math. Soc. 52 (1975), ) demonstrate how each type of the generalized p,-stirling numbers of the first second kind arises as a special case of these p, -rook numbers. Résumé. Remmel et Wachs, dans (EJC 11 (2004), #R84), ont présenté deu façons naturelles pour définir les p, -analogues des nombres de Stirling généralisés, des première et deuième sortes, S 1 (α, β, r) et S 2 (α, β, r), introduits par Hsu et Shiue (Adv. App. Math 20 (1998), ). Dans cet article, nous présentons un model théoriue des mouvements de la tour pour chaue type des p, -analogues basé sur une paire de eu paramétrisés par les entiers non-négatifs α, β, et r. Ainsi, la tour attaue les cases sur son propre eu et celles de l autre eu. Pour chacun des modèles, nous donnons une formule analogue à celle du produit de Goldman, Joichi et White (Proc. Amer. Math. Soc. 52 (1975), ) et démontrons comment chaue type de p, -analogues des nombres de Stirling généralisés des première et deuième sortes forment un cas spécial de nombres p,-analogues pour les mouvements de la tour. 1. Introduction In [11], Remmel Wachs presented two natural ways to give p, -analogues of Hsu Shiue s generalized Stirling numbers of the first second kind [7], respectively denoted S 1 (α, β, r) S 2 (α, β, r) for 0 k n, defined by n (1.1) ( α) ( (n 1)α) = S 1 (α, β, r)( r)( r β) ( r (k 1)β), (1.2) ( β) ( (n 1)β) = n S 2 (α, β, r)( + r)( + r α) ( + r (k 1)α). From these definitions, one can clearly see that S 1 (α, β, r) = S 2 (β, α, r). Moreover, we find that S 1 (1, 0, 0) = s S 2 (1, 0, 0) = S where s S respectively denote the classical Stirling numbers of the first second kind. By setting S 1 (α, β, r) = S1 (α, β, r) S2 (α, β, r) = S2 (α, β, r) 2000 Mathematics Subect Classification. Primary 05A10, 05A15, 05A30; Secondary 05E15. Key words phrases. generalized stirling numbers, p, -analogues, rook numbers, bipartite boards. This research was completed as part of a larger proect on rook theory with Jeffrey B. Remmel, Department of Mathematics, University of California, San Diego.

2 K. Briggs replacing by t r in euation (1.1) by t in euation (1.2), Remmel Wachs obtained the following pair of euations: n (1.3) (t r)(t r α) (t r (n 1)α) = S 1 (α, β, r)t(t β) (t (k 1)β), (1.4) t(t β) (t (n 1)β) = n S(α, 2 β, r)(t r)(t r α) (t r (k 1)α). Replacing (t γ) by two distinctly natural p, -analogues, Remmel Wachs then defined their two types of p, -analogues of S 1 (α, β, r) S2 (α, β, r). The p, -analogue of any real number γ is defined by [γ] p, = pγ γ p, so that when γ = n is a nonnegative integer, [n] p, = n 1 +p n 2 + p n 2 +p n 1. Then, the p, -analogues of n! ( n k) are naturally defined by [n]p,! = [n] p, [n 1] p, [1] p, [ ] n [n] p,! = k [k] p,![n k] p,!. p, For their type-i (p, )-analogues of S 1 (α, β, r) S2 (α, β, r), Remmel Wachs replaced (t γ) by ([t] p, [γ] p, ) in (1.3) (1.4). That is, they defined S 1,p, respectively by the following euations: (1.5) (1.6) (α, β, r) S2,p, (α, β, r) for 0 k n ([t] p, [r] p, )([t] p, [r + α] p, ) ([t] p, [r + (n 1)α] p, ) n = S 1,p, (α, β, r)([t] p,)([t] p, [β] p, ) ([t] p, [(k 1)β] p, ) ([t] p, [β] p, ) ([t] p, [(n 1)β] p, ) n = S 2,p, (α, β, r)([t] p, [r] p, )([t] p, [r + α] p, ) ([t] p, [r + (k 1)α] p, ). Moreover, they proved that when 0 k n, the S 1,p, (α, β, r) S2,p, (α, β, r) defined according to euations (1.5) (1.6) satisfy the following recursions: (1.7) S 1,p, 0,0 (α, β, r) = 1 S1,p, (α, β, r) = 0 if k < 0 or k > n (1.8) S 1,p, n+1,k (α, β, r) = S1,p, 1 (α, β, r) + ([kβ] p, [nα + r] p, )S 1,p, (α, β, r), (1.9) S 2,p, 0,0 (α, β, r) = 1 S2,p, (α, β, r) = 0 if k < 0 or k > n (1.10) S 2,p, n+1,k (α, β, r) = S2,p, 1 (α, β, r) + ([kα + r] p, [nβ] p, )S 2,p, (α, β, r). For their type-ii (p, )-analogues of S 1 (α, β, r) S2 (α, β, r), Remmel Wachs replaced (t γ) 1,p, 2,p, by [t γ] p, in (1.3) (1.4). That is, they defined S (α, β, r) S (α, β, r) for 0 k n by the following euations: n (1.11) [t r] p, [t r α] p, [t r (n 1)α] p, = S 1,p, (α, β, r)[t] p,[t β] p, [t (k 1)β] p, (1.12) [t] p, [t β] p, [t (k 1)β] p, = n S 2,p, (α, β, r)[t r] p,[t r α] p, [t r (k 1)α] p,.

3 A ROOK THEORY MODEL FOR THE GENERALIZED p, -STIRLING NUMBERS b a a a a b Figure 1. A placement in (N F) 2 3 (B BIP(1, 0, 3, 3, 5, 6, 7, 8)). S 1,p, They further proved that when 0 k n, the euations (1.11) (1.12) satisfy the following recursions: 2,p, (α, β, r) S (α, β, r) defined according to (1.13) S1,p, n+1,k (α, β, r) = (k 1)β nα r S1,p, 1 (α, β, r) + pt kβ [kβ nα r] S1,p, p, (α, β, r), with initial conditions S 1,p, 0,0 1,p, (α, β, r) = 1 S (α, β, r) = 0 if k < 0 or k > n, (1.14) S2,p, n+1,k (α, β, r) = r+(k 1)α nβ S2,p, 1 (α, β, r) + pt r kα [kα + r nβ] S2,p, p, (α, β, r), S 2,p, 0,0 2,p, with initial conditions (α, β, r) = 1 S (α, β, r) = 0 if k < 0 or k > n. Remmel Wachs gave rook theory interpretations to c i, (p, ) = ( 1)n k S 1,p, (, 0, i) Si, (p, ) = S 2,p, i, 1,p, (, 0, i) as well as ci, (p, ) = ( 1)n k S1,p, (, 0, i) S (p, ) = S (, 0, i) where i, are nonnegative integers. Moreover, they were able to give combinatorial proofs of certain product formulas involving these polynomials. In this paper, we provide a generalization of their results by giving combinatorial interpretations to S i,p, i,p, (α, β, r) S (α, β, r) when α, β r are integers i {1, 2}, we give combinatorial proofs to the product formulas that Remmel Wachs did not provide. S 1,p, 2. A Rook Theoretic Model for S 1,p, (α, β,r) S2,p,(α, β,r) In this section, we give a rook theoretic model to interpret the type-i generalized p, -Stirling numbers (α, β, r) S2,p, (α, β, r). The boards in our model are constructed as follows. Given any two finite seuences of nonnegative integers {a 1, a 2,..., a n } {b 1, b 2,...,b n }, we construct the bipartite board B BIP (a 1, b 1, a 2, b 2,..., a n, b n ) whose column heights from left to right are a 1, b 1, a 2, b 2,..., a n, b n. We will call the collection of columns whose heights are a 1, a 2,...,a n, the Premier-columns (P-columns), the collection of columns whose heights are b 1, b 2,..., b n the Secondary-columns (S-columns). For eample, from the seuences {1, 3, 5, 7} {0, 3, 6, 8}, we obtain the board B BIP (1, 0, 3, 3, 5, 6, 7, 8) which is illustrated in Figure 1 with the P-columns given in white the S-columns shaded in gray. For any bipartite board B = B BIP (a 1, b 1, a 2, b 2,...,a n, b n ), a rook r placed in a P-column (resp. S-column) of B is said to -attack the cells in the P-columns of B that are strictly to the right of r in the first rows that are weakly above r (resp. in the first rows beginning with row 1) that are not -attacked by any other rook that lies in a column to the left of r. Then, a placement P of rooks in B BIP (a 1, b 1, a 2, b 2,..., a n, b n ) is called -nonattacking if no rook in P is -attacked by any rook in P to its left there is at most one -attacking rook per column pair {a i, b i } for each 1 i n. We let (N F) k (B) denote the set of all placements of k -nonattacking rooks in B. A placement in (N F) 2 3 (B BIP(1, 0, 3, 3, 5, 6, 7, 8)) is illustrated in Figure 1. As usual, rooks are denoted in the figure by an. In this eample, the leftmost rook in B is placed in row 2 of column a 1 2-attacks the cells in rows 2 3 of columns a 3 a 4. These cells 2-attacked by the leftmost rook contain an a in Figure 1. The second rook from the left in row 3 of column b 3 2-attacks the cells in rows 1 4 of column a 4. These cells 2-attacked by this second rook contain a b in Figure 1. The final rook of the placement is in row 6 of column a 4. Since there are no P-columns to the right of a 4, this rook does not -attack any cells in the board.

4 K. Briggs p p p p p Figure 2. The p, -weight of P (N F) 2 3 (B BIP(1, 0, 3, 3, 5, 6, 7, 8)) as contributed to r k (B, p, ). For a nonnegative integer, we say that a board B = B BIP (a 1, b 1, a 2, b 2,...,a n, b n ) is a -attacking bipartite board if 0 a 1 a 2 a n, 0 b 1 b 2 b n, for all placements of rooks in B, there are a sufficient number of cells in the P-columns of B for each rook to -attack. By this definition, note that in the case when b 1 = = b n = 0, the board B BIP (a 1, 0, a 2, 0,, a n, 0) is a -attacking bipartite board provided that for all 1 i < n, a i 0 implies that a i+1 a i + 1. However, in the case when b i 0 for some 1 i < n, B is a -attacking bipartite board provided that a +1 a + for all > i 1. In Figure 1, the board B BIP (1, 0, 3, 3, 5, 6, 7, 8) is a 2-attacking bipartite board. Suppose that P (N F) k (B) set n S (P) = the number of rooks of P placed in an S-column, A B = the number of non-attacked cells in B directly above some rook in P, B B = the number of non-attacked cells in B directly below some rook in P, w p,,b (P) = ( 1)nS(P) AB p BB. The type-i p, -rook numbers, denoted r k (B, p, ), are defined by (2.1) r k (B, p, ) = P (N F) k (B) w p,,b (P). Here in what follows, we will place a in the cells -attacked by rooks in a given placement P, a in the cells that contribute a factor of to w p,,b (P), a p in the cells that contribute a factor of p to w p,,b (P). As illustrated in Figure 2 for P (N F)2 3 (B BIP(1, 0, 3, 3, 5, 6, 7, 8)), w p,,b (P) = ( 1)2 9 p 5. Our first result is a p, -analogue of Goldman, Joichi, White s product formula [5]. Theorem 2.1. Let B = B BIP (s, b 1, s +, b 2,..., s + (n 1), b n ). Then n (2.2) r n k (B, p, )([] p, [s] p, )([] p, [s + ] p, ) ([] p, [s + (k 1)] p, ) = ([] p, [b 1 ] p, )([] p, [b 2 ] p, ) ([] p, [b n ] p, ). Proof. Given B = B BIP (s, b 1, s+, b 2,...,s+(n 1), b n ), we let B (,) be the board obtained from B by adoining a single column of height + s + (i 1) beneath the column pair {a i, b i } for each 1 i n. Here we call the line separating B from the adoined rows the bar, the first rows below the bar in B (,) the -adoined rows the last s + (n 1) rows in B (,) below the bar the -adoined rows. Further, we will call the collection of cells in the column pair {a i, b i } together with the + s + (i 1) adoined cells below it the ith oined column. The augmented board B (,) is illustrated in Figure 3. For a given board B, placements of -attacking rooks placed above the bar in B (,) will -attack the same cells above the bar as described above. Additionally, any -attacking rook r placed above the bar will attack all of the cells below it in its oined column as well as the first rows in the -adoined rows strictly to the right of r not attacked by any rook to the left. A rook that is placed in one of the -adoined rows will attack all of the cells directly above it in the board B as well as the cells directly below it in the -adoined rows. A rook that is placed in a -adoined row will attack the cells directly above it in the -adoined

5 A ROOK THEORY MODEL FOR THE GENERALIZED p, -STIRLING NUMBERS.. b 1 b 2 b n s adoined s adoined.. Figure 3. The board B (,) (s, b 1, s +, b 2,...,s + (n 1), b n ) rows in the board B. The cells attacked by rooks of a placement in B (,) (1, 0, 3, 3, 5, 6, 7, 8) have been illustrated in Figure 4. Let (N F) n(b (,) ) be the set of all placements of n rooks in B (,) such that no two rooks lie in the same oined column, no rook in B (,) is -attacked by any rook in B (,) to its left. Thus, the placement in Figure 4 is in (N F) 2 4 (B BIP(1, 0, 3, 3, 5, 6, 7, 8) (,2) ). Then for positive integers, the identity in (2.2) arises from two ways of counting (2.3) N = w p,,b (,) (P), where w p,,b (,) (P) is defined as with P (N F) n(b (,) ) w p,,b (,) (P) = ( 1) ns+n AB (,) p B B(,), n S (P) = the number of rooks of P placed in an S-column, n (P) = the number of rooks of P placed in a -adoined row, A B(,) = the number of non-attacked cells in B (,) directly above some rook in P, B B(,) = the number of non-attacked cells in B (,) directly below some rook in P. First we note that each placement P (N F) n(b (,) ) can be obtained by placing eactly one rook in each of the oined columns of B (,) proceeding from left to right. In the first oined column, the rook can be placed in either the first P-column, the first S-column, below the bar in the -adoined rows, or in the -adoined rows. It is easy to see that the contribution of the first oined column to N by placing the rook in the ith row from the top in the first P-column is i 1 p s i for a total contribution of [s] p, to N. Likewise, the contribution of the first oined column to N by placing the rook in the ith row from the top in the first

6 K. Briggs Figure 4. A placement in the board B (,2) (B(1, 0, 3, 3, 5, 6, 7, 8)). S-column is i 1 p b1 i for a total contribution of [b 1 ] p, to N. Using the same analysis, we find that when the rook is placed below the bar in the -adoined rows, the contribution of the first oined column to N is [] p, while the total contribution is [s] p, from the placements in the -adoined rows. Therefore, the total contribution of the first oined column to N is [] p, [b 1 ] p,. We now argue that regardless of the placement of the rook in the first oined column, the contributions from the P-column the -adoined of the second adoined column will cancel. To see this, first consider the case when a rook in the first oined column had been placed in B. Such a rook would attack eactly cells in the P-columns as well as cells in each row of the -adoined columns weakly to the right of the rook. In this case, the contribution from the second P-column is [s] p, while the contribution from the second -adoined column is [s] p,. On the other h, if the rook in the first oined column had been placed below the bar, then the contribution from the second P-column is [s + ] p, while the contribution from the second -adoined column is [s + ] p,. To this end, we can argue as above, that the contribution of the second adoined column to N is [] p, [b 2 ] p,. Continuing in this way, we find that the total contribution of all n adoined columns to N is ([] p, [b 1 ] p, )([] p, [b 2 ] p, ) ([] p, [b n ] p, ). Now suppose that a placement Q of n k rooks is fied in B. Then a placement P (N F) n(b (,) ) can be obtained from Q by placing the remaining k rooks below the bar. As prescribed, each of the n k rooks in B will attack cells in the -adoined rows in the columns weakly to the right of each rook. As such, there will be places in the -adoined rows s + (i 1) places in the -adoined rows in which to place the ith rook below the bar from the left, for 1 i k. Therefore, the placement of the k rooks below the bar will contribute a factor of ([] p, [s] p, )([] p, [s + ] p, ) ([] p, [s + (k 1)] p, ) to N. Furthermore, each rook placed below the bar will attack the cells above it in B implying that w p,,b (Q) = w p,,b (P B).

7 A ROOK THEORY MODEL FOR THE GENERALIZED p, -STIRLING NUMBERS Thus, N = = = n Q (N F) n k (B) P (N F) n (B (,) ) P B=Q n Q (N F) n k (B) w p,,b (,) (P) w p,,b (P B)([] p, [s] p, )([] p, [s + ] p, ) ([] p, [s + (k 1)] p, ) n r n k (B, p, )([] p, [s] p, )([] p, [s + ] p, ) ([] p, [s + (k 1)] p, ). We are now in a position to give combinatorial interpretations to S 1,p, (α, β, r) S2,p, (α, β, r) defined by (1.5) (1.6). To begin, let, y, µ ν be nonnegative integers let Bµ,ν,n,y denote the bipartite board B BIP (, y, + µ, y + ν, + 2µ, y + 2ν,..., + (n 1)µ, y + (n 1)ν). Then, Theorem 2.2. If n k are nonnegative integers for which 0 < k < n, then (2.4) (α, β, r) = rβ n k (B0,r, p, ) (2.5) S 2,p, (α, β, r) = rα n k(b r,0, p, ). Proof. We begin by noting that the identities in (2.4) (2.5) can be proved by showing that the S 1,p, p, -rook numbers r β n k (B0,r that r α n k (Br,0 For n = 0, B,y µ,ν,0, p, ) satisfy the same recursion as S1,p,(α, β, r) given in (1.7) (1.8), p, ) satisfy the same recursion as S2,p, (α, β, r) given in (1.9) (1.10). =. So, it immediately follows from our definition that r β 0 (B0,r β,α,0, p, ) = 1 rα 0 (B r,0 α,β,0, p, ) = 1. Clearly, r β n k (B0,r, p, ) = 0 rα n k (Br,0, p, ) = 0 if k > n or k < 0 since both (N F)β n k (B0,r ) (N F) α n k (Br,0 ) are empty if k > n or k < 0. Therefore, to verify the eualities in (2.4) (2.5), it remains to show that for all n 1 0 k n, (2.6) r β n+1 k (B0,r, p, ) = rβ n (k 1) (B0,r, p, ) + ([kβ] p, [nα + r] p, )r β n k (B0,r, p, ) (2.7) rn+1 k(b α r,0, p, ) = rα n (k 1) (Br,0, p, ) + ([kα + r] p, [nβ] p, )rn k(b α r,0, p, ). To prove (2.6), we note that the set of elements in (N F) β n+1 k (B0,r ) can be partitioned into the sets No, P Last, S Last where No consists of the placements of (N F) β n+1 k (B0,r ) with no rook in the column pair {a n+1, b n+1 }, P Last consists of the placements of (N F) β n+1 k (B0,r ) with a rook in the P-column a n+1, S Last consists of the placements of (N F) β n+1 k (B0,r ) with a rook in the S-column b n+1. Then, r β n+1 k (B0,r, p, ) = β w P (N F) β n+1 k (B0,r ) = P No w β (P) + P P Last (P) w β (P) + P S Last w β (P). It is easy to see that a placement in P No has n (k 1) rooks to the left of the column pair {a n+1, b n+1 }. Thus, w β (P) = w β (Q) where Q (N F) β n (k 1) (B0,r ) is the placement that would result in eliminating the last pair of columns {a n+1, b n+1 } from B 0,r. Therefore, P No w β (P) = Q (N F) β n (k 1) (B0,r β w ) (Q) = r β n (k 1) (B0,r, p, ).

8 K. Briggs To compute P P Last wβ (P), we first observe that each Q (N F) β n k (B0,r ) can be etended to kβ placements in P Last by placing an additional rook in a non-attacked cell of column a n+1. This follows since each of the n k rooks of a fied Q (N F) β n k (B0,r ) attacks β cells in column a n+1 leaving nβ (n k)β = kβ non-attacked cells in column a n+1 in which to place the additional rook. Net, we note that n S (P) = n S (Q). So, if the additional rook is placed in the ith non-attacked cell from the top, then the weight of the corresponding placement P i is i 1 p kβ i w β (Q). Therefore, P P Last w β (P) = Q (N F) β n k (B0,r ) = [kβ] p, r β (B0,r, p, ). (p kβ 1 + p kβ kβ 2 p + kβ 1 ) w β (Q) Finally, we observe that each rook of P S Last attacks β cells of column a n+1 but no cells of column b n+1. Accordingly, P S Last wβ (P) could be computed by etending each Q (N F) β n k (B0,r ) to nα + r distinct placements in (N F) β n k+1 (B0,r ) by placing an additional rook in any of the nα + r non-attacked cells of column b n+1. For such a placement P i obtained by placing the additional rook in the ith non-attacked cell from the top, we note that n S (P i ) = 1 + n S (Q) conseuently w β (P i ) = i 1 p nα+r+i w β (Q). Therefore, it follows that P S Last = w β (P) Q (N F) β n k (B0,r ) = [nα + r] p, r β (B0,r, p, ). (p nα+r 1 + p nα+r nα+r 2 p + nα+r 1 ) w β In the same way, we prove (2.7) by partitioning (N F) α n+1 k (Br,0 ) into the sets No, P Last, S Last where No consists of the placements of (N F) α n+1 k (Br,0 ) with no rook in the column pair {a n+1, b n+1 }, P Last consists of the placements of (N F) α n+1 k (Br,0 (Q) ) with a rook in the P-column a n+1, S Last consists of the placements of (N F) α n+1 k (Br,0 ) with a rook in the S-column b n+1. The recursion in (2.7) will follow by showing that rn+1 k α (B0,r, p, ) = α w p,,b r,0 P (N F) α n+1 k (Br,0 ) = P No Again, it is easy to see that P No w α p,,b r,0 (P) + P P Last (P) w α p,,b r,0 w α (P) = r p,,b r,0 n (k 1) α (Br,0, p, ). (P) + P S Last w α (P). p,,b r,0 To compute P P Last wα (P), we observe that each fied placement Q (N F) α p,,b r,0 n k (Br,0 ) can be etended to kα + r distinct placements in (N F) α n k (Br,0 ) by placing an additional rook in one of the nα + r α(n k) = kα + r non-attacked cells of column a n+1. If the additional rook is placed in the ith non-attacked cells from the top of column a n+1, then the weight of the corresponding placement P i is i 1 p kα+r i w α (Q). It follows that p,,b r,0 P P Last w α p,,b r,0 (P) = Q (N F) α n k (Br,0 ) (p kα+r 1 + p kα+r kα+r 2 p + kα+r 1 ) w α p,,b r,0 = [kα + r] p, r(b α r,0, p, ). (Q)

9 A ROOK THEORY MODEL FOR THE GENERALIZED p, -STIRLING NUMBERS As above, we observe that each rook of P S Last attacks α cells of column a n+1 but no cells of column b n+1. Therefore, P S Last wα (P) could be computed by etending each Q (N F) α p,,b r,0 n k (Br,0 ) to nβ distinct placements in (N F) α n k+1 (Br,0 ) by placing an additional rook in any of the nβ non-attacked cells of column b n+1. For such a placement P i obtained by placing the additional rook in the ith non-attacked cell from the top, we note that n S (P i ) = 1 + n S (Q) thus w α (P s ) = i 1 p nβ+i w α (Q). To this end, P S Last w α (P) = p,,b r,0 Q (N F) α n k (Br,0 ) = [nβ] p, r α (Br,0, p, ). (p nβ 1 + p nβ nβ 2 p + nβ 1 ) w α (Q) p,,b r,0 We end this section by noting that as a conseuence of Theorems , our single model described above yields a combinatorial interpretation to both (1.5) (1.6). In particular, (1.5) is obtained from (2.2) by setting s = 0, = β, b i = r + (i 1)α. Likewise, setting s = r, = α, b i = (i 1)β in (2.2) produces (1.5). 3. A Rook Theoretic Model for S 1,p, (α, β,r) S 2,p, (α, β,r) To define the second type of p, -rook numbers, let B be a -attacking bipartite board suppose P (N F) k (B). Assume additionally that the k rooks are in the column pairs {a i, b i } with labels 1 c 1 < < c k n that there are i non-attacked cells in the column containing the rook among the pair {a ci, b ci } for 1 i k. Setting a B = the number of non-attacked cells in B directly above some rook in P, b B = the number of non-attacked cells in B directly below some rook in P, ε B = the number of non-attacked cells in a P-column of an {a i, b i } pair containing no rook, we define the type-ii p, -rook numbers, denoted r k (B, p, ), by (3.1) r k (B, p, ) = (b1+ +bn) P (N F) k (B) ( 1) ns εb(p)+ab(p) p bb(p)+kt (1+2+ +k). The following result gives the generalized product formula for the type-ii p, -rook numbers. Theorem 3.1. Let B = B BIP (a 1, b 1, a 2, b 2,...,a n, b n ) be a -attacking bipartite board. Then for each nonnegative integer n, n (3.2) r n k (B, p, )[t] p,[t ] p, [t (k 1)] p, = n bi( ) [t + a i (i 1)] p, p t+ai (i 1) bi [b i ] p,. i=1 The proof of Theorem 3.1 is similar to that of Theorem 2.1. The idea is to consider all placements of n -attacking rooks in the board B t (a 1, b 1, a 2, b 2,..., a n, b n ) which is obtained from B by adoining t rows below the n P-columns, labeled from bottom to top by 1, 2,...,t. The board B t (a 1, b 1, a 2, b 2,...,a n, b n ) is illustrated in Figure 5. Here, rooks placed in B will -attack in the P-columns as usual, while a rook that is placed in row i below a P-column will -attack the cells to the right in the first rows weakly above it in the list of rows i, i + 1,...,t, 1,...,i 1 that have not been -attacked by a rook from the left. Such a placement of n rooks is illustrated in Figure 5 with = 2. It can also be shown that the type-ii p, -rook numbers on specific -attacking bipartite boards satisfy the same recursions as the type-ii Stirling numbers of the first second kind. To see this, we first note that (3.3) [kβ nα r] p, = nα r ( [kβ] p, p kβ nα r [nα + r] )

10 K. Briggs t Figure 5. The board B t (a 1, b 1, a 2, b 2,..., a n, b n ) (3.4) [kα + r nβ] p, = nβ ( [kα + r] p, p kα+r nβ [nβ] ). Then, substituting the identity (3.3) into the recursion (1.13) (3.4) into (1.14) yields the following three term recursions: S 1,p, n+1,k (3.5) (α, β, r) = (k 1)β nα r S1,p, 1 (α, β, r) + pt kβ nα r [kβ] S1,p, p, (α, β, r) p t nα r nα r 1,p, [nα + r] p, S (α, β, r). S 2,p, n+1,k (3.6) (α, β, r) = r+(k 1)α nβ S2,p, 1 (α, β, r) + pt r kα nβ [kα + r] S2,p, p, (α, β, r) p t nβ nβ 2,p, [nβ] p, S (α, β, r). As in the proof of Theorem 2.2, we can show that the rook numbers r β n k (B0,r ) rα n k (Br,0 ) satisfy the respective recursions in (3.5) (3.5) by again partitioning the set of placements (N F) β n+1 (B0,r ) (N F) α n+1(b r,0 ) into No, P Last, S Last. We summarize these results in the following: Theorem 3.2. If n k are nonnegative integers for which 0 < k < n, then (3.7) S2,p, (3.8) S 1,p, (α, β, r) = rβ n k (B0,r ) (α, β, r) = rα n k(b r,0 ). As a conseuence of Theorems , this single rook theoretic model yields a combinatorial interpretation for the identities given in (1.11) (1.12). To see this, we observe that bi( [t + a i (i 1)] p, p t+ai (i 1) bi [b i ] p, ) = [t + ai (i 1) b i ] p,. Then from (3.2), (1.11) is obtained by setting = β, a i = (i 1)β, b i = r + (i 1)α as is (1.12) by setting = α, a i = r + (i 1)α, b i = (i 1)β, replacing t with t r.

11 A ROOK THEORY MODEL FOR THE GENERALIZED p, -STIRLING NUMBERS 4. Directions While our models have provided a rook theoretic interpretation for both types of p, -analogues of the generalized Stirling numbers of the first second kind, their product formulas recursions, we have yet to produce analogues of the orthogonality relations given by Hsu Shiue [7] for arbitrary parameters α, β, r: n n S 1 (α, β, r)s 2 k,i(α, β, r) = S 2 (α, β, r)s 1 k,i(α, β, r) = χ(i = n). k=i k=i Although, Remmel Wachs gave direct combinatorial interpretations of the following p, -analogues of the orthogonality relations n n k=r k=r S 2,p, (, 0, i)s1,p, k,r (, 0, i) = χ(r = n) p (n k+1 2 ) S2,p, (, 0, i)(p)(k 2) p ir ik S1,p, k,r (, 0, i) = χ(r = n), they did not provide the p, -orthogonality relations for arbitrary parameters α, β, r. We will pursue this problem in a subseuent paper. References [1] K. S. Briggs, A Combinatorial Interpretation for p, -Hit Numbers, Discrete Mathematics, submitted. [2] K. S. Briggs, Q-analogues p, -analogues of rook numbers hit numbers their etensions, Ph.D. thesis, University of California, San Diego (2003). [3] K. S. Briggs J. B. Remmel, A p, -analogue of a Formula of Frobenius, Electron. J. Comb. 10 (2003), #R9. [4] A. M. Garsia J. B. Remmel, Q-Counting Rook Configurations a Formula of Frobenius, J. Combin. Theory Ser. A 41, [5] J. R. Goldman, J. T. Joichi, D. E. White, Rook Theory I. Rook euivalence of Ferrers boards, Proc. Amer. Math Soc. 52 (1975), [6] H. G. Gould, The -Stirling numbers of the first second kinds, Duke Math. J. 28 (1961), [7] L. C. Hsu P. J. S. Shiue, A Unified Approach to Generalized Stirling Numbers, Advances in Applied Mathematics 20 (1998), [8] A. de Médicis P. Lerou, A unified combinatorial approach for -( p, -)Stirling numbers, J. Statist. Plann. Inference 34 (1993), [9] A. de Médicis P. Lerou, Generalized Stirling Numbers, Convolution Formulae p, -analogues, Can. J. Math. 47 (1995), [10] S. C. Milne, Restricted growth functions, rank row matchings of partition lattices, -Stirling numbers, Adv. in Math. 43 (1982), [11] J. B. Remmel M. Wachs, Rook Theory, Generalized Stirling Numbers (p, )-analogues, Electron. J. Comb. 11 (2004), #R84. [12] I. Kaplansky J. Riordan, The problem of rooks its applications, Duke Math. J. 13 (1946), [13] M. Wachs D. White, p, -Stirling numbers set partition statistics, J. Combin. Theory Ser. A 56 (1991), [14] M. Wachs, σ-resticted growth functions p, -Stirling numbers, J. Combin. Theory Ser. A 68 (1994), Department of Mathematics, Clayton State University, USA, address: karenbriggs@clayton.edu