Forml Power Series nd Algebric Combintorics Séries Formelles et Combintoire Algébriue Sn Diego, Cliforni 006 Generl Augmented Rook Bords & Product Formuls Brin K Miceli Abstrct There re number of so-clled fctoriztion theorems for rook polynomils tht hve ppered in the literture For exmple, Goldmn, Joichi, nd White [6] showed tht for ny Ferrers bord B = F(b, b,, b n), ny n (x + b i (i )) = r k (B)(x) () where r k (B) is the k-th rook number of B nd (x) k = x(x ) (x (k )) is the usul flling fctoril polynomil Similr formuls where r k (B) is replced by some pproprite generliztion of rook numbers nd (x) k is replced by polynomils like (x) k,j = x(x + j) (x + j(k )) or (x) k,j = x(x j) (x j(k )) cn be found in the work of Goldmn nd Hglund [5], Remmel nd Wchs [], Hglund nd Remmel [7], nd Briggs nd Remmel [] We shll cll such formuls generlized product formuls The min gol of this pper is to develop new rook theory setting where we cn give uniform combintoril proof of generlized product formul which includes ll the cses referred to bove Tht is, given ny two seuences of non-negtive integers, B = (b,, b n) nd A = (,, n), nd two sign functions sgn, sgn : {,, n} {, }, we shll define rook theory setting nd pproprite generliztion of rook numbers rk A (B, sgn,sgn) such tht ny n (x + sgn(i)b i ) = rk A (B, sgn,sgn) Y j (x + ( sgn(s) s)) Thus, for exmple, we obtin combintoril interprettions of the connection coefficients between ny two bses of the polynomil ring Q[x] of the form {(x) k,j } k 0 or {(x) k,j } k 0 We lso find -nlogues nd (p, )-nlogues of the bove formuls j= s= Résumé Le but principl de cet rticle est de développer une nouvelle théorie rook dns luelle nous pouvons fournir des preuves combintoires uniformes d une formule de produit générlisée ui inclut toutes les cs cités ci-dessus C est-à-dire, se donnnt deux suites uelconues de nombres entiers positifs, B = (b,, b n) et A = (,, n), et deux fonctions de signes sgn,sgn : {,, n} {, }, nous définissons une théorie rook insi u une générlistion ppropriée des nombres rook rk A (B, sgn, sgn) tel ue ny n (x + sgn(i)b i ) = rk A (B, sgn,sgn) Y j (x + ( sgn(s) s)) Donc, pr exemple, nous obtenons une interpréttion combintoire des coefficients de connexion entre deux bses de l nneu des polynômes Q[x] de l forme {(x) k,j } k 0 ou {(x) k,j } k 0 Nous trouvons ussi des -nlogues et des (p, )-nlogues de ces formules j= s= Introduction Let N = {,,, } denote the set of nturl numbers For ny positive integer, we will set [] := {,,, } We will sy tht B n = [n] [n] is n n by n rry of sures (like chess bord), which we 000 Mthemtics Subject Clssifiction Primry 05A0; Secondry 05A5 Key words nd phrses lgebric combintorics, rook theory, enumertion, inverses This pper dpted from section of the doctorl thesis of the uthor with thnks to the direction nd ssistnce provided by the thesis dvisor, Jeffrey Remmel
B K Miceli cll cells The cells of B n will be numbered from left to right nd bottom to top with the numbers from [n], nd we will refer to the cell in the i th row nd j th column of B n s the (i, j) cell of B n Any subset of B n is clled rook bord If B is bord in B n with column heights b, b,, b n reding from left to right, with 0 b i n for ech i, then we will write B = F(b, b,, b n ) B n In the specil cse tht 0 b b b n n, we will sy tht B = F(b, b,, b n ) is Ferrers bord Given bord B = F(b, b,, b n ), there re three sets of numbers we cn ssocite with B, nmely, the rook, file, nd hit numbers of B The rook number, r k (B), is the number of plcements of k rooks in the bord B so tht no two rooks lie in the sme row or column The file number, f k (B), is the number of plcements of k rooks in the bord B so tht no two rooks lie in the sme column but where we llow ny given row to contin more thn one rook Given permuttion σ = σ σ σ n in the symmetric group S n, we shll identify σ with the plcement P σ = {(, σ ), (, σ ),, (n, σ n )} Then the hit number, h k (B), is the number of σ S n such tht the plcement P σ intersects the bord in exctly k cells All of these numbers hve been studied extensively by combintorilists Here re three fundmentl identities involving these numbers Define (x) m = x(x ) (x (m )) nd (x) m = x(x + ) (x + (m )) Then () h k (B)x k = r k (B)(n k)!(x ) k, () () (x + b i (i )) = (x + b i ) = r (B)(x) k, nd f (B)x k Identity () is due to Kplnsky nd Riordn [8] nd holds for ny bord B B n Identity () holds for ll Ferrers bords B = F(b,,b n ) nd is due to Goldmn, Joichi, nd White [6] Identity () is due to Grsi nd Remmel [4] nd holds for ll bords of the form B = F(b,, b n ) Formuls () nd () re exmples of wht we shll cll product formuls in rook theory We note tht in the specil cse where B = B n := F(0,,,, n ), Eutions () nd () become (4) x n = r (B n )(x) k nd (5) (x) n = f (B n )x k This shows tht r (B n ) = S n,k, where S n,k is the Stirling number of the second kind, nd ( ) f (B n ) = s n,k, where s n,k is the Stirling number of the first kind, nd thus, we obtin rook theory interprettions for the Stirling numbers of the first nd second kind There re nturl -nlogues of formuls (), (), nd () Tht is, define [n] = ++ + n = n We then define -nlogues of the fctorils nd flling fctorils by [n]! = [n] [n ] [] [] nd [x] m = [x] [x ] [x (m )], Grsi nd Remmel [4] defined -nlogues of the hit numbers, h k (B, ), -nlogues of the rook numbers, r k (B, ), nd -nlogues of file numbers, f k (B, ), for Ferrers bords B so tht the following hold: (6) h k (B, )x = r (B, )[k]!x k ( x k+ ) ( x n ), (7) [x + b i (i )] = r (B, )[x] k, nd (8) [x + b i ] = f (B, )([x] ) k
GENERAL AUGMENTED ROOK BOARDS Finlly, we should mention tht there re lso (p, )-nlogues of such formuls (see Wchs nd White [], Briggs nd Remmel [], nd Briggs []) In recent yers, number of reserchers hve developed new rook theory models which give rise to new clsses of product formuls For exmple, Hglund nd Remmel [7] developed rook theory model where the nlogue of the the rook number m k (B) counts prtil mtchings in the complete grph K n They defined n nlogue of Ferrers bord F(, n ) where n n 0 nd where the nonzero entries in (,, n ) re strictly decresing, nd, in their setting, they proved the following identity, (9) (x + n i i + ) = n n m k (F)x(x )(x 4) (x (n (k ))) Remmel nd Wchs [] defined more restricted clss of rook numbers, r j k (B), in their j-ttcking rook model nd proved tht for Ferrers bords B = F(b,,b n ), where b i+ b i j if b i 0, (0) (x + b i j(i )) = r j (B)x(x j)(x j) (x (k )j) Goldmn nd Hglund [5] developed n i-cretion rook theory model nd proved tht for Ferrers bords one hs the following identity, () (x + b i + j(i )) = r (i) (B)x(x + (i )) (x + (k )(i )) j= In ll of these new models, the uthors proved -nlogues nd or (p, )-nlogues of their product formuls A Generl Product Formul Suppose we re given ny two seuences of nturl numbers: B = {b i } n, A = { i} n Nn Define A i = + + + i, the i th prtil sum of the i s, nd let B = F(b, b,, b n ) be rook bord We will lso define two functions, sgn nd sgn, such tht sgn, sgn : [n] {, +} Our gol is to define rook theory model with n pproprite notion of the rook numbers rk A (B, sgn, sgn) such tht the following product formul holds: () (x + sgn(i)(b i )) = rk A (B, sgn, sgn) j=(x + s j sgn(s)( s )) We will refer to Eution () s the generl product formul nd the number rk A (B, sgn, sgn) s the kth ugmented rook number of B with respect to A, sgn, nd sgn Specil Cses of the Generl Product Formul We first wish to consider the cse where sgn(i) = + nd sgn(i) = for every i n In this cse we will set Thus, we wnt to prove Eution (): () (x + b i ) = r A k (B, sgn, sgn) = ra k (B) rk A (B)(x A )(x A ) (x A ) k=o To do this, we first construct n ugmented rook bord, B A = F(b + A, b + A,, b n + A n ) In B A, the cells in the i-th column re (, i),, (b i + + + i, i) reding from bottom to top We shll refer to the cells (, i),,(b i, i) s the b i prt of column i, the cells (b i +, i),,(b i + A i, i) s the A i prt of column i, nd, for ech s i, the cells (b i + + s +, i),,(b i + + + s, i) s the s prt of column i where by convention = 0 We cll the prt of the bord B A which corresponds to the A i s the ugmented prt of B A We now consider rook plcements in B A with t most one rook in ech column We define the following cncelltion rule: rook r plced in column j of B A will cncel, in ech column to its right, ll of the cells which lie in the i prt of tht column where i is the highest subscript j such tht the j prt of tht column hs not been cnceled by rook to the left of r For exmple, in Figure,
B K Miceli 4 * * b b b b 4 Figure B A, with B = F(,,, 4) nd A = (,,, ), nd plcement of two rook in B A where B = F(,,, 4) nd A = (,,, ), the rook in the first column cncels the cells in the prt of the second column, the prt of the third column, nd the 4 prt of the fourth column (those cells which contin ) The rook in the third column cncels the cells in the prt the fourth column (those cells which contin ) We then define rk A(B) to be the number of wys of plcing k such rooks in BA so tht no rook lies in cell which is cnceled by rook to its left We cn now construct generl ugmented rook bord, Bx A, defined by the seuences B = {b i } n nd A = { i } n nd some nonnegtive integer x The bord BA x will be the bord B A (the ugmented prt of B A will here be referred to s the upper ugmented prt of Bx A ), with x rows ppended below, clled the x-prt nd then mirror imge of the ugmented prt of B A below tht, clled the lower ugmented prt of Bx A In the lower ugmented prt, we number the cells in i-th column with (, i),,(b i + A i, i) reding from top to bottom nd we define the s prt of the i-th column of the lower ugmented bord to consist of the cells ( + + s +, i),,( + + s, i) We sy tht the bord B A is seprted from the x-prt by the high br nd the x-prt is seprted from the lower ugmented prt by the low br An illustrtion of this type of bord with B = F(,,, 4), A = (,,, ), nd x = 4 cn be seen in the left side of Figure In order to define proper rook plcement in the bord Bx A, we mke the rule tht exctly one rook must be plced in every column of Bx A When plcing rooks in BA x, we will define the following cncelltion rules: () A rook plced bove the high br in the j th column of Bx A will cncel ll of the cells in columns j +, j +,,n, both in the upper nd lower ugmented prts, which belong to the i prt of the column where i is lrgest j such tht cells in the j prt of the column re not cnceled by rook to their left () Rooks plced below the high br do not cncel ny cells An exmple of rook plcement in these bords cn be seen in the right side of Figure In this plcement, the rook plced in the first column is plced bove the high br, thus it cncels in the columns to its right those cells contined in the i prt of highest subscript in both the upper nd lower ugmented prts (denoted by ) The rook plced in the second column is plced below the the high br so tht it cncels nothing The rook plced in the third column is gin plced bove the high br so tht it cncels s does the rook plced in the first column (denoted by ), nd the lst rook my be plced in ny vilble cell We will now prove two lemms in order to prove Eution () Lemm If there re b j +A m cells to plce rook bove the high br in column j, then there re A m cells below the low br to plce rook in column j Proof: By how we define our cncelltion, block of cells from i gets cnceled bove the high br if nd only if block of cells from i gets cnceled below the low br Lemm If k rooks re plced bove the high br in Bx A, then the column heights of the uncnceled cells in the lower ugmented prt of Bx A, when red from left to right, re A, A,, A Proof: Suppose the first rook bove the high br is plced in the j th column The columns below the low br which lie to the left of column j hve heights A, A,, A j Now, the rook tht ws plced in the
GENERAL AUGMENTED ROOK BOARDS b b b 4 b 4 high br * * x prt low br * * 4 Figure B A x, with B = F(,,, 4), A = (,,, ), nd x = 4, nd plcement of rooks in B A x the j th column will cncel ll the cells in the j+ prt of the (j + ) st column, ll the cells in the j+ prt of the (j + ) nd column, etc Thus fter this cncelltion, the heights of the columns below the low br into which rook my be plced re A, A,,A j, A j, A n Now suppose tht the leftmost rook to the right to column j is in column k Then the rook in column k will cncel ll the cells in k prt of the (k + ) st column, ll the cells in the k+ prt of the (k + ) nd column, etc Thus fter this second cncelltion, the heights of the columns below the low br into which rook my be plced re A, A,, A j, A j, A k, A k,, A n We cn continue this type of resoning to show tht if there re k rooks re plced bove the high br in Bx A, then the column heights of the uncnceled cells in the lower ugmented prt of Bx A, when red from left to right, re A, A,, A We re now in position to prove () We shll show tht () is the result of computing the sum S of the weights of ll plcements of n rooks in Bx A in two different wys, where we define the weight of the rooks plced bove the low br to be +, the weight of the rooks plced below the low br to be, nd the weight of ny plcement to be the product of the weights of the rooks in the plcement If we first plce the rooks strting with the leftmost column nd working to the right, then we cn see tht in the first column there re exctly x + b + cells in which to plce the first rook, where the corresponds to plcing the rook in either the upper or lower ugmented prt of the st column Since ll of the rooks bove the high br hve weight + nd ll the rooks plced below the low br hve weight, it is esy to see tht the possible plcements of rooks in the first column contributes fctor of x+b + +( ) = x+b to S When we consider the possible plcements of rook in the second column, we hve two cses Cse I: Suppose the rook tht the st column ws plced below the high br Then nothing ws cnceled in the second column so we cn plce rook in ny cell of the second column Thus there re totl
B K Miceli x + b + ( + ) wys to plce the rook in the second column in this cse However, given our weighting of the rooks, we see tht the possible plcements of rooks in the second column contributes fctor of x + b + ( + ) + ( ) = x + b to S Cse II: If the rook in the first column ws plced bove the high br, then the cells corresponding to prt in both the upper nd lower ugmented prts of the nd column re cnceled Thus in this cse, there re x+b + cells left to plce the rook in the second column However, given our weighting of rooks, the possible plcements of rooks in the second column contributes fctor of x + b + ( ) + = x + b to S in this cse In generl, suppose we re plcing rook in the j th column tht does not hve rook bove the high br reding from left to right Assume tht we hve plced s rooks bove the high br nd t rooks below the high br in the first j columns Then by Lemm, we hve, x + b j + (A j s ) choices s to where to plce the rook in tht column Agin, due to our weighting, it is esy to see tht possible plcement of rooks contributes fctor of x + b j + A j s + ( A j s ) = x + b j to S It follows tht S = n (x + b i) The second wy of counting this sum of the weights of ll the rook plcements in Bx A is to orgnize the plcements by how mny rooks lie bove the high br Suppose tht we plce k rooks bove the high br nd then wish to extend tht to plcement in the entire bord The number of wys of plcing the k rooks bove the high br is given by rk A (B) For ny such plcement of k rooks bove the high br, we re left with n k columns in which to plce rooks below the high br We consider the plcement of the remining rooks in these vilble columns strting with the leftmost one nd working right By Lemm, the number of wys we cn do this will be (x + A )(x + A ) (x + A )) However, these plcements come with weighting of (x + ( A )(x + ( A ) (x + ( A )) since the cells below the low br hve weight Thus the sum of the weights of the set of plcements in Bx A with k rooks bove the high br is rk A(B)(x A )(x A ) (x A )) Summing over ll possible k gives us the RHS of () Now suppose we chnge the weights which re ssigned to rooks in Bx A by declring tht the weight of rook plced in the upper ugmented prt is nd ll other rooks hve weight + Agin the weight of the plcement is the product of the weights of the rooks in the plcement This weighting corresponds to the cse where sgn(i) = + nd sgn(i) = + for every i n We will define r k A (B) to be the weighting of ll plcements of k rooks in B A with this newly ssigned weight, nd this yields n eution which is nlogous to Eution (), nmely, () (x + b i ) = r k A (B)(x + A )(x + A ) (x + A ) k=o Proof of Eution (): This proof follows exctly the proof of Eution () with the weights from the upper nd lower ugmented prts switched We cn see tht these two specil cses encpsulte ll of the product formuls stted in the Introduction Next we sketch proof for the generl product formul () The Generl Product Formul We hve now shown how to generte our generl product formul in the specil cses where the functions sgn nd sgn re certin constnt functions; however, the proofs of Eutions () nd () do not depend on sgn nd sgn being constnt Rther, the proofs depend only on the condition tht, for ech column j, if the cells corresponding to the i prt of the upper ugmented prt in column j re weighted with ω( i ), then the cells corresponding to the i prt in the lower ugmented prt in column j must be weighted with -ω( i ) Moreover, the proofs do not depend on the weighting of rooks plced in the cells of the cells in the b i prt of column i in B A Thus, if we define rk A (B, sgn, sgn) to be the weight of ll plcements of k rooks in the bord B A, with ech rook in the b i prt of column i hving weight sgn(i) nd ech rook in the i prt of ny column below the low br hving weight sgn(i), nd the weight of ny plcement to be the product of the rooks in tht plcement, then we cn show tht Eution () is the result of computing the sum S of the weights of ll plcements of n rooks in Bx A exctly s in the proofs of Eutions () nd ()
GENERAL AUGMENTED ROOK BOARDS Q-Anlogues of Generl Product Formuls In this section, we shll describe how one cn derive -nlogues of some of the generl product formuls described in Section We do this by -counting rook plcements considered in Section To simplify our nottion, we shll use the convention tht for ny negtive integer x, [x] := [ x ] If we set A k = k sgn(i) i, then we cn prove the following -nlogue of Eution : () ([x] + sgn(i)[b i ] ) = rk A (B, sgn, sgn, ) ([x] + [A s ] ) For ech cell c in the bord B A, we let below B A(c) denote the number of cells tht lie directly below c in its prt Tht is, if c is cell in the ugmented prt of B A, then below B A(c) is the number of cells below c in the ugmented prt of B A nd if c is not in the ugmented prt of B A, then below B A(c) is just the number of cells below c in B We my then extend this definition to the bord Bx A by defining below B A x (c) to be the number of cells below given cell c in Bx A in its prt To ech cell c in the bord Bx A we will ssign -weight, ω (c) Given plcement P in Bx A, we will define the -weight of tht plcement to be ω (P) = r P ω (r), where ω (r) = ω (c) if the rook r is in cell c First, we define ω (c) = below B x A(c) if c is in the x-prt of the bord Next we set ω (c) = sgn(i) below B x A(c) if c is in the i th column of the bord B For the lower ugmented prt of the bord, the definition of ω (c) is slightly more involved Suppose we re t the k th column of the lower ugmented prt of Bx A, which hs column height + + + k Recll tht we lbeled the cells in k-th column of the lower ugmented bord from top to bottom with the pirs (, k), (, k),,( + + k, k) Then, for i, we set ω ((i, k)) = sgn() i Now, ssume by induction tht we hve ssigned weights to the cells (, k), (, k),,( + + i, k) so tht + + i j= ω ((j, k)) = [A i ] Then we will lbel the cells ( + + i +, k),,( + + i + i+, k) in the following mnner: () Cse I: A i 0 () If A i A i+, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai+,, Ai+, respectively (b) If 0 A i+ A i, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai,, A i+, respectively (c) If A i+ < 0, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai,,,,,,, Ai+, respectively () Cse II: A i < 0 () If A i A i+, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai +,, Ai+, respectively (b) If 0 A i+ A i, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai,, Ai+, respectively (c) If A i+ > 0, then we ssign the -weight of the cells ( + + i +, k),,( + + i + i+, k) to be Ai, Ai,,,,,,, Ai+, respectively Finlly, in order to ssign the -weights to the k th column of the upper ugmented prt of Bx A, we will simply tke the weights tht we ssigned to the lower ugmented prt of the k th column, flip them upside down nd multiply them ll by - An exmple of this weighting cn be seen in Figure, where the -number displyed in ech cell of the digrm corresponds to the -weight rook plced in tht cell would be given For exmple, we cn see tht the -weights ssigned to the lower ugmented prt of the fourth column, red from top to bottom re:,,,,, The weights in the upper ugmented prt of the sme column re, when red from bottom to top:,,,,,, which is the previous seuence with every element multiplied by - Now we cn prove Eution similr to the wy we proved Eution in the previous section Tht is, Eution results by computing the sum S of the -weights of ll plcements of n rooks in Bx A in two different wys Specil Cses of the Generl Q-Anlogue Formul Now consider the specil cses where sgn nd sgn re the constnt functions or + In this cse, it is esy to see tht s=
B K Miceli Figure A -nlogue of the rook plcement in Figure with sgn(i) = + for i =,,, 4 nd sgn(i) = + for i =,, 4 sgn(i) = for i = Here ech cell is lbeled with the -weight tht rook plced in tht cell would be given (i) if rook in is the b i prt of column i of B A, then its -weight will be sgn(i) below B A x (c), (ii) if rook in is the A i prt of column i of B A, then its -weight will be sgn(i) below B A x (c), (iii) if rook in is the x prt of column i of B A, then its -weight will be below B A x (c), nd (iv) the -weights of cell in the lower ugmented prt of the bord is just the -weight of its mirror imge in the upper ugmented prt of the bord multiplied by In this cse () becomes () ([x] + sgn(i)[b i ] ) = rk A (B, sgn, sgn, ) ([x] + sgn(s)[a s ] ) It turns out tht by slightly modifying our -counting of rook plcements, we cn prove nlogues of () where we replce [x] [c] by [x c] or [x] + [c] by [x + c] Cse I: sgn(i) = sgn(i) = For x, c N with x > c, we hve tht [x] [c] = c [x c] Thus () becomes s= () bi [x b i ] = rk A (B, sgn, sgn, ) As [x A s ] s=
GENERAL AUGMENTED ROOK BOARDS It is then esy to see tht if we replce rk A(B, ) with ˆrA k (B, ) by we obtin the following form of Formul : (4) ([x b i ] ) = ˆr k A (B, ) := (A+A+ +A ) (b + +b n) rk A (B, ), ˆr k A (B, )([x A ] )([x A ] ) ([x A ] ) k=o Cse II: sgn(i) =, sgn(i) = + For x, c N we hve tht [x] + x [c] = [x + c] Thus if we wnt to replce [x] + [A i ] by [x + A i ] = [x] + x [A i ], then we should weight ech rook tht lies in upper ugmented prt of B A by n extr fctor of x This mens tht when we consider plcements in Bx A, then we must lso weight ech rook tht lies in the lower ugmented prt of BA x with n extr fctor of x so tht for ny given column the weights of possible plcements in the lower nd upper ugmented prts cncel ech other s in the proofs in Section Thus we define ˆr k A (B, ) to be the sum of the -weight over ll plcements of k rooks in B A where ech rook plced in the ugmented prt receiving n extr fctor of x Then it is esy to see tht () becomes (5) ([x] [b i ] ) = ˆr k A (B, )[x + A ] [x + A ] Finlly we replce ˆr k A(B, ) by new -rook number, r k A(B, ) where r k A(B, ) := (b+ +bn) rk A (B, ) In doing this, we obtin the following formul: (6) ([x b i ] ) = r A k (B, )([x + A ] )([x + A ] ) ([x + A ] ) We cn lso use methods similr to the ones used in Cses I nd II, to prove the following product formuls for pproprite choices of r A k (B, ) nd ra k (B, ) Cse III: sgn(i) = +, sgn(i) = (7) ([x + b i ] ) = r A k (B, )([x A ] )([x A ] ) ([x A ] ) k=o 4 Cse IV: sgn(i) = sgn(i) = + (8) ([x + b i ] ) = r A k (B, )([x + A ] )([x + A ] ) ([x + A ] ) k=o 4 (P, Q)-Anlogues of Generl Product Formuls For ny n N we define [n] p, = p n +p n + + n p+ n, nd we gin use the convention tht for negtive integer x, [x] p, := [ x ] p, Then we cn give combintoril interprettion of the following (p, )-nlogue formul: (4) ([x] p, + sgn(i)[b i ] p, ) = rk A (B, sgn, sgn, p, ) ([x] p, + [A s ] p, ) Agin, we will ssign weight to ech cell c of the bord Bx A, which we will cll the (p, )-weight of c, nd this will be denoted by ω p, (c) We will lso define the sttistic bove B A x (c), for ny cell c in the bord Bx A, to be the number of cells tht lie bove c in its prt, tht is, if c is in the x-prt of the bord, then bove B A x (c) is the number of cells tht lie bove c in the x-prt For plcement P of rooks in Bx A, we will let the (p, )-weight of P be ω p, (P) = r P ω p,(r), where ω p, (r) = ω p, (c) if the rook r is plced in cell c Now, we cn (p, )-weight the cells of Bx A in the following mnner: () If c is in the x-prt of the bord, then ω p, (c) = p bove B A x (c) below B A x (c) () If c is in the i th column of the bord B, then ω p, (c) = sgn(i)p bove B A x (c) below B A x (c) s=
B K Miceli () If c is in the k th column of the lower ugmented prt of the bord, then we will set ω(, k) = [A ] p, We will then set ω( + + i +, k) = [A i+ ] p, [A i ] p, nd ω(j, k) = 0 otherwise (4) If c is in the k th column of the upper ugmented prt of the bord, then weights will be ssigned, from bottom to top, s they were in the lower ugmented prt, with ll of the weights multiplied by - We note tht this type of weighting is more complicted thn our -weighting since now cell cn receive (p, )-weight which is polynomil in p nd rther thn just plus or minus power of Moreover, there re mny other choices we could mke for the weights, but none of them reduce to the -weight when p = However, in certin specil cses, we cn ssign more nturl (p, )-weight which is consistent with some of the (p, )-nlogues of product formuls tht hve ppered in the literture, but we shll not consider these types of results in this pper We cn now prove Eution 4 in the exct sme wy tht we proved Eution 5 Conclusion nd Perspectives We hve given rook theory interprettion of the product formul (x + sgn(i)b i ) = rk A (B, sgn, sgn) j (x + ( sgn(s) s )), nd this interprettion cn be used to obtin identities studied by Goldmn nd Hglund [5], Remmel nd Wchs [], Hglund nd Remmel [7], nd Briggs nd Remmel [] We lso hve - nd (p, )- nlogues of this generl product formul One ppliction of this new theory is in finding the inverses of connection coefficients for different bses of Q[x] [9] If we define the functions (x) k, = x(x + )(x + ) (x + (k )) nd (x) k,b = x(x b)(x b) (x (k )b), then for ny N, the sets {(x) n, } n 0 nd {(x) n, } n 0 will both form bses of Q[x] Thus, there exist numbers C n,k (b, ) nd C n,k (, b ) such tht (5) (x) n,b = nd (5) (x) n, = j= C n,k (b, )(x) k, C n,k (, b )(x) k,b From liner lgebr it is known tht C n,k (b, ) = C n,k (, b ), tht is to sy, (5) s= C n,j (, b )C j,k (b, ) = χ(n = k) j=k However, this result my be obtined from our rook theory model Given the numbers, b N we will define B = (0, b, b,, (n )b) nd A = (0,,,,) By now defining sgn(i) = nd sgn(i) = + we see tht C n,k (b, ) = rk A(B, sgn, sgn) nd C n,k(, b ) = rk B (A, sgn, sgn) We cn now write eutions (5) nd (5) s (54) (x) n,b = nd (55) (x) n, = In prticulr, we now hve tht rk A (B, sgn, sgn)(x) k, rk B (A, sgn, sgn)(x) k,b
GENERAL AUGMENTED ROOK BOARDS (56) j=k r A (B, sgn, sgn)rb j k (A, sgn, sgn) = χ(n = k), nd we hve completely combintoril proof of this fct bsed solely on involutions on rook plcements in n ugmented rook bord setting We cn give similr combintoril proofs for ll the possible choice of nd in the coefficient C n,k (b, ) For exmple, we cn find combintoril interprettions of the inverses of the numbers C n,k (, b ) nd C n,k (, b ) which stisfy the eutions (57) (x) n, = nd (58) (x) n, = C n,k (, b )(x) k,b C n,k (, b )(x) k,b Another ppliction of our rook theory model reltes to the numbers S p(x) n,k (59) S p(x) n+,k = Sp(x) n,k + p(k)sp(x) n,k, defined in [0] by where p(x) is ny polynomil with nonnegtive integer coefficients nd with initil conditions S p(x) 0,0 = nd S p(x) n,k = 0 whenever n < 0, k < 0, or n < k We cll such numbers poly-stirling numbers of the second kind [0] Then, for exmple, in the specil cse where p(x) = x m, we cn use n extension of the theory of generl ugmented rook bords to give combintoril proof of the formul (50) (x n ) m = Sn,k xm j k (x m (j ) m ) Finlly, we should note tht theory of hit numbers corresponding to the rook theory for our generlized product formuls hs yet to be developed References [] K S Briggs, -Anlogues nd p, -Anlogues of Rook Numbers nd Hit Numbers nd Their Extensions, Thesis, UC, Sn Diego (005) [] K S Briggs nd J B Remmel, A p, -nlogue of Formul of Frobenius, Electronic Journl of Combintorics 0() (00), #R9 [] K S Briggs nd J B Remmel, m-rook Numbers nd Generliztion of Formul of Frobenius for C m S n, Electronic Journl of Combintorics, submitted [4] A M Grsi nd J B Remmel, Q-Counting Rook Configurtions nd Formul of Frobenius, J Combin Theory Ser A 4 (986), 46-75 [5] J Goldmn nd J Hglund Generlized Rook Polynomils, J Combin Theory Ser A 9 (000), 509-50 [6] J R Goldmn, J T Joichi, nd D E White, Rook Theory I Rook euivlence of Ferrers bords, Proc Amer Mth Soc 5 (975), 485-49 [7] J Hglund nd J B Remmel, Rook Theory for Perfect Mtchings, Advnces in Applied Mthemtics 7 (00), 48-48 [8] I Kplnsky nd J Riordn, The problem of rooks nd its pplictions, Duke Mth J (946), 59-68 [9] J Liese, B K Miceli, nd J B Remmel, The Combintorics of the Connection Coefficients of Generlized Rising nd Flling Fctoril Polynomils, In Preprtion [0] B K Miceli, TBD, Thesis, UC, Sn Diego (006) [] J B Remmel nd M Wchs, Generlized p, -Stirling numbers, privte communiction [] M Wchs nd D White, p,-stirling numbers nd set prtition sttistics, J Combin Theory Ser A 56 (99), 7-46 Deprtment of Mthemtics, University of Cliforni, Sn Diego, L Joll, CA, USA, 909-0 E-mil ddress: bmiceli@mthucsdedu URL: http://wwwmthucsdedu/~bmiceli