The digraph drop polynomial

Transcription

1 The digraph drop polyomial Fa Chug Ro Graham Abstract For a weighted directed graph (or digraph, for short), deoted by D = (V, E, w), we defie a two-variable polyomial B D (x, y), called the drop polyomial of D, which depeds itimately o the cycle structure of the permutatios o vertices of D. This polyomial geeralizes several other digraph polyomials which have bee studied i the literature previously, such as the biomial drop polyomial for posets [2], the biomial drop polyomial for digraphs [7], the path/cycle cover polyomial for digraphs [5] ad the matrix cover polyomial [6]. We show that B D (x, y) satisfies a reductio/cotractio recurrece as well as the (somewhat mysterious) reciprocity relatio B D(x, y) = ( 1) B D ( x y, y), where D has vertices ad D = (V, E, w) is the digraph formed from D by defiig w(e) = 1 w(e) for each edge e E. 1 Itroductio Suppose P = (V, ) is a partially-ordered set o a set V of cardiality. I [2], a polyomial B P (x) called the biomial drop polyomial for P was itroduced which was defied as follows. If π : V V is a permutatio o V, we say that π has a drop at u if π(u) = v ad v u. Let drop(π) deote the umber of drops that π has. The defie B P (x) = ( ) x + drop(π), π where π rus over all! permutatios o V. The polyomial B P (x) turs out to have may coectios to a variety of subjects i combiatorics. For 1

2 example, suppose we defie the icomparability graph Ic(P ) of P to be the graph o V where {u, v} is a edge of Ic(p) if ad oly if u ad v are icomparable i P. Let χ Ic(P ) (x) deote the chromatic polyomial of Ic(P ) (e.g., see [12]). The for ay positive iteger b, B P (b) = χ Ic(P ) (b). Further, B P ( 1) is equal to the umber of permutatios π o V which have o drops. This is also equal to the umber of acyclic orietatios of Ic(P ) by a classic theorem of Staley [12]. I the case that P is a chai, i.e., a liearly-ordered set, the Ic(P ) has o edges ad we have B P (x) = x = ( ) x + k k k where k deotes the usual Euleria umber. This is usually kow as Worpitzky s idetity (see [9]). There is a atural digraph D = (V, E) associated to a poset P = (V, ). Namely, the vertex set for D is V. For u, v V we have a directed edge (u, v) E if ad oly v u i P. (1) Thus, oe might ask if (1) could be exteded to more geeral digraphs. I fact, this ca be doe as follows for ay give digraph D. Suppose π : V V is a permutatio o V. Let us say that π has a drop at u if (u, π(u)) E. Deote by drop(π) to the the umber of drops that π has. The followig polyomial B D (x), called the biomial drop polyomial for D, which geeralizes (1), was studied i [7]: B D (x) = ( ) x + drop(π) π where V = ad π rus over all! permutatios o V. This polyomial ca be applied to a more geeral class of digraphs, called weighted digraphs. A weighted digraph D = (V, E, w) is a digraph o the vertex set V with a edge set E V V i which each edge e E is assiged a real-valued weight w(e) (possibly 0). I this case, if π is a permutatio o V, we geeralize the (2) 2

3 defiitio of a drop(π) by defiig drop(π) = u V e=(u,π(u)) w(e). (3) We ca the exted the defiitio of B D (x) to weighted digraphs as follows: B D (x) = ( ) x + drop(π) (4) π where, as before, the vertex set V of D has size. Here, we iterpret the biomial coefficiet i the sum as ( ) x + α =!(x + α) =!(x + α)(x + α 1)... (x + α + 1) where we use the fallig factorial otatio z = z(z 1)(z 2)... (z +1). It was show i [7] that B D (x) has a umber of iterestig properties, such as a deletio/cotractio recurrece similar i form to the well-kow deletio/cotractio recurrece for the Tutte polyomial (see [14]). The polyomial we will cosider i this paper is a geeralizatio of B D (x) to a 2-variable polyomial B D (x, y), called the drop polyomial for D. However, before defiig B D (x, y), we first metio a related digraph polyomial. This is the so-called (path/cycle) cover polyomial C D (x, y) for D (cf. [5]). For a simple digraph D = (V, E), that is, oe i which all edge weights are 1, we ormally thik of a edge e = (u, v) E as represeted by a directed arc goig from u to v. A path/cycle cover C of E is a collectio of vertex-disjoit paths ad cycles which cover all the vertices i V, where we cosider a sigle vertex to be a path of legth zero, ad a loop (u, u) to be a cycle of legth oe. Let c D (i, j) deote the umber of path/cycle covers of E cosistig of i paths ad j cycles. The (path/cycle) cover polyomial for D is defied by C D (x, y) = i,j c D (i, j)x i y j. (5) 3

4 The cover polyomial C D (x, y) was geeralized i [6] to weighted digraphs D = (V, E, w) as follows. For a path/cycle cover C of D, let p(c) deote the umber of paths i C, ad let cyc(c) deote the umber of cycles i C. By the weight w(c) of C we mea the product of all the weights of the edges i C. The the geeralizatio of C D (x, y) to weighted digraphs is: C D (x, y) = C path/cycle cover x p(c) y cyc(c) w(c). (6) It was show i [6] that i additio to a deletio/cotractio recurrece, C D (x, y) satisfies a umber of other iterestig properties such as: (i) If D 1 = (V 1, E 1, w 1 ) ad D 2 = (V 2, E 2, w 2 ) are vertex disjoit digraphs ad the digraph D is formed from D 1 ad D 2 by joiig every v 1 V 1 to every v 2 V 2 by a edge (v 1, v 2 ) of weight 1, the C D (x, y) = C D1 (x, y)c D2 (x, y); (ii) If D = (V, E, w) deotes the weighted digraph formed from D = (V, E, w) by subtractig the weight of each edge e E from 1, i.e., w(e) = 1 w(e), ad V =, the we have the surprisig reciprocity formula: C D(x, y) = ( 1) C D ( x y, y). (7) This paper is orgaized as follows: I Sectio 2, we will give the defiitio of drop polyomial B D (x, y) for a weighted digraph D = (V, E, w) i which each edge e E is assiged some weight w(e). We will ordiarily assume that w(e) is a real umber (possibly 0) although all of our results are valid if w(e) just lies i some commutative rig with idetity. Basically, B D (x, y) is a sum over all permutatios π o V where each term of the sum depeds o drop(π) ad the cycle structure of π restricted to D. I Sectio 3, we discuss a umber of useful facts about B D (x, y). A alterative defiitio of B D (x, y) is give i Sectio 4. The drop polyomials B D (x, y) are similar to but differet from the path-cycle polyomials C D (x, y). We will show i Sectio 6 that they coicide if D is a simple digraph. However, for geeral digraphs, they satisfy 4

5 differet deletio/cotractio rules as see i Sectio 5. Nevertheless, for both polyomials the same reciprocity theorems hold as show i Sectio 7. The drop polyomials have direct coectios with permutatios o the vertex set ad i tur lead to itriguig questios cocerig eumeratio problems for permutatios ad partial permutatios, some of which will be metioed i Sectio 8. 2 The drop polyomial B D (x, y) As usual, we start with a weighted digraph D = (V, E, w) o a set V of size (possibly havig some edges of weight 0.) For a permutatio π : V V we defie E(π) to be the set of edges e = (u, v) such that π(u) = v. Thus, E(π) cosists of paths ad cycles. I this case, loops, i.e., edges of the form (u, u), are still cosidered to be cycles (of legth 1). Let P (π) deoted the set of edges i paths i E(π) ad let Cyc(π) deote the set of cycles i E(π). Also, let cyc(π) deote Cyc(π). More geerally, for ay subset S E, we ca defie cyc(s) i the obvious way, i.e., as the umber of cycles formed by the edges i S. Further, we defie the weight w(s) by: w(s) = w(e). (8) e S We fially come to the defiitio of the drop polyomial B D (x, y): B D (x, y) = ( ) x + w(s) ( 1) E(π)\S (y 1) cyc(π) cyc(s) y cyc(s) π P (π) S E(π) (9) where π rages over all permutatio of the vertex set V of D ad V =. For example, suppose that D = I(), the digraph o vertices with o edges. The it is easy to see that i this case, B I() (x, y) = x, where we recall the fallig factorial otatio z = z(z 1)(z 2)... (z + 1). Note that if we substitute y = 1 i (9), the B D (x, y) reduces to the biomial drop polyomial B D (x) i (4). For i this case, the oly term i 5

6 the ier sum i (9) which does t vaish is for the choice S = E(π), ad i this case, ( ) x + w(s) B D (x, 1) = π ( ) x + drop(π) = π = B D (x) which is (4), as desired. 3 Addig weight 0 edges. For the geeral weighted digraphs D = (V, E, w) we have bee cosiderig up to ow, E is some subset of V V with various weights assiged to the edges i E. However, there will be some advatages i beig able to assume that E is all of V V, where pairs (u, v) which were ot origially i E are replaced by edges e = (u, v) with weight w(e) = 0. Of course, the dowside of doig this is that ow the sums ivolved i evaluatig (9) may have may more terms tha before. Oe upside is that equivalet forms of (9) become simpler (cf. Sectio 4). We ext show that this modificatio of D (replacig o-edges by edges of weight 0) does ot chage the value of B D (x, y). We first show that B D (x, y) is uchaged if a sigle pair (u, v) / E is replaced by a edge e = (u, v) with w(e) = 0. Equivaletly, we show that removig a edge of weight 0 does ot chage B D (x, y). Theorem 1. Suppose D = (V, E, w) is a weighted digraph, ad e 0 E with w(e 0 ) = 0. Form the digraph D = (V, E, w) by deletig the edge e 0, i.e., E = E \ e 0. The B D (x, y) = B D (x, y). 6

7 Proof. Whe referrig to D, we will use E (π), cyc (π), etc., to deote the correspodig parameters for D. Thus, E = E 1. From the defiitio i (9), we have B D (x, y) = π P (π) S E (π) ( ) ( 1) E (π)\s x + w(s ) (y 1) cyc (π) cyc (S ) y cyc (S ) The overall pla is to show that B D (x, y) reduces to B D (x, y) whe w(e 0 ) = 0. To do this, we are goig to expad the terms of B D (x, y) for each π ad show how i each case they either cacel each other out, or correspod to uique terms i B D (x, y). So cosider some fixed permutatio π o V. Case (i). e 0 / E(π). I this case, for each choice of S i the sum for B D we ca choose S = S i the correspodig term i the sum for B D. Sice i this case, E (π) = E(π), c (π) = c(π), c (S ) = c(s), etc., the correspodig terms i B D ad B D are equal. Case (ii). e 0 P (π). I this case, we will always have e 0 S, sice P (π) S. Hece, we ca write S = S e 0 for some S with P (π) S E (π). Now we have E (π) = E(π) 1, S = S 1, E (π)\s = E(π)\S, cyc (π) = cyc(π) ad cyc (S ) = cyc(s). Sice w(s) = w(s ) + w(e 0 ), the whe w(e 0 ) is 0, the correspodig terms i B D ad B D are equal i this case as well. Case (iii). e 0 C(π). Thus, e 0 belogs to some cycle C = {e 0, e 1, e 2,..., e r } i D (where r = 0 is allowed). Let C deote the set C \ {e 0 }. There are ow two possibilities for S: (a) C S. I this case, we will fid a uique mate S for S as follows. If e 0 / S the set S = S e 0 ; if e 0 S the set S = S \ e 0. We will examie the sum of the two terms ( ) x + w(s) ( 1) E(π)\S (y 1) cyc(π) cyc(s) y cyc(s) (10) P (π) S E(π) 7

8 ad E(π) S E(π) ( 1) E(π)\S ( ) x + w(s ) (y 1) cyc(π) cyc(s ) y cyc(s ) (11) i B D. Suppose S = S \ e 0 (the other case is symmetric). The S = S + 1, cyc(s ) = cyc(s) ad w(s) = w(s ) + w(e 0 ) = w(s ). Thus, ( 1) E(π)\S = ( 1) E(π)\S so that the sum of the two terms (10) ad (11) is zero. (b) C S. I this case we ca write S = C T where T E(π) \ C. We have P (π) = P (π) C ad we set S = C T. Now we have cyc(s) = cyc(s ) + 1. Thus, sice S = S + 1 the addig (10) ad (11), we get ( ) x + w(s) ( 1) E(π)\S (y 1) cyc(π) cyc(s) y cyc(s) P (π) S E(π)) ( ) + ( 1) E(π)\S x + w(s ) (y 1) cyc(π) cyc(s ) y cyc(s ) P (π) S E(π) ( ) = ( 1) E(π)\S x + w(s ) + w(e 0 ) (y 1) cyc(π) cyc(s ) 1 y cyc(s )+1 P (π) S E(π) ( ) + ( 1) E(π)\S x + w(s ) (y 1) cyc(π) cyc(s ) y cyc(s ) P (π) S E(π) ( ) = ( 1) E(π)\S x + w(s ) (y 1) cyc(π) cyc(s ) 1 y cyc(s ) ((y 1) y) P (π) S E(π) ( ) = ( 1) E(π)\S x + w(s ) (y 1) cyc(π) cyc(s ) 1 y cyc(s ). P (π) S E(π) Now, takig S = S i B D ad otig that cyc (π) = cyc(π) + 1 ad E(π) = E (π) + 1, we see that the term i this last sum becomes ( ) ( 1) E (π)\s x + w(s ) (y 1) cyc (π) c (S) y cyc (S) (12) 8

9 This is exactly the term correspodig to the choice of S, with P (π) S E (π), i B D. The precedig argumets show there is a bijectio betwee the terms of B D ad the (o-cacelig) terms of B D. Thus, we have proved B D (x, y) = B D (x, y). Corollary 1. Suppose D = (V, E, w) is a weighted digraph, ad D = (V, E, w) is formed by addig a edge e = (u, v) to E with w(e) = 0 for every pair (u, v) / E. The B D (x, y) = B D (x, y). Proof. Just apply Theorem 1 recursively. 4 A alterative form for B D (x, y). Suppose D = (V, E, w) is a weighted digraph where E = V V (so that there may be may weight 0 edges). A partial permutatio σ o V is a ijective mappig σ : U V for some subset U V. There are several ways to represet σ. Oe is by the familiar two-lie otatio: σ = u 1 u 2... u k σ(u 1 ) σ(u 2 )... σ(u k ) where u i s deote distict vertices of D. Here, we assume that U = k ad we will say that σ has size σ = k. Aother way to represet σ is to idetify it with the correspodig etries i a associated matrix M = M(D) idexed by elemets of V. Here, the etry M(u i, σ(u i )) of M is just the weight w(e i ) of the edge e i = (u i, σ(u i )). If S(σ) deotes the set of edges formed by σ, the it is clear that S(σ) cosists of vertex disjoit paths ad cycles. Sice we assume E = V V, we ca abuse otatio slightly by just sayig that S = S(σ) is a partial permutatio, sice 9

10 specifyig the etries i the matrix M automatically determies the actual partial permutatio σ. Deote by w(s) deote the sum of the weights of the edges i S ad let cyc(s) the umber of cycles i D formed by the edges i S. Also, let P P (V ) deote the set of all partial permutatios o V. We the have the followig alterative expressio for B D (x, y). Theorem 2. For ay weighted digraph D, we have B D (x, y) = ( ) x + w(s) (1 y) S y cyc(s). (13) S P P (V ) Proof. From Theorem 1, we may assume E = V V by addig edges of weight 0 while the drop polyomial B D (x, y) remais uchaged. Cosequetly, for ay permutatio π : V V, we have P (π) =. Hece, we ca iterchage the order of summatio i (9) to obtai: B D (x, y) = ( ) x + w(s) ( 1) E(π)\S (y 1) cyc(π) cyc(s) y cyc(s) π S E(π) = ( ) x + w(s) ( 1) E(π)\S (y 1) cyc(π) cyc(s) y cyc(s) S P P (V ) π where π rages over all permutatios of V for which S E(π) so that S rages over all partial permutatios. The umber of π with k cycles is give by [ k ], a Stirlig umber of the first kid (see [9]). More geerally, if S = s, the π has s free blocks from which to form k cyc(s) cycles. Thus, there are just [ S k cyc(s)] such π with S E(π). A basic idetity for Stirlig umbers of the first kid is the followig (see [9]): k [ m k ] z k = z m (14) where z m deotes the risig factorial z m = z(z + 1)... (z + m 1). Hece 10

11 our expressio for B D (x, y) becomes B D (x, y) = ( ) x + w(s) [ ] S ( 1) S (y 1) k cyc(s) y cyc(s) k cyc(s) S P P (V ) k = ( ) x + w(s) ( 1) S (y 1) S y cyc(s) S P P (V ) = ( ) x + w(s) (1 y) S y cyc(s) S P P (V ) which is (13). This completes the proof of Theorem 2. As a example, cosider the digraph D = (V, E, w) where V = [], E = [] [] ad w(e) = 0 for all e E. Sice i this case the reduced digraph is just I(), a digraph with o edges, ad we have see that B I() (x, y) = x. Thus, by (13), we get the iterestig idetity: Corollary 2. S P P (V ) (1 y) S y cyc(s) =!. It does t appear obvious (to us) how to prove this directly. 5 A reductio/cotractio rule for B D (x, y) I order to maipulate B D (x, y), we will first eed to defie the operatios of reductio ad cotractio i a digraph D = (V, E, w) (see Figures 1 ad 2). First, suppose e = (u, v) is a o-loop edge of D (so u v). The e-reduced digraph D = (V, E, w ) has the same set of vertices ad edges as D, with the oly chage beig the weight of e is reduced by 1, i.e., w (e) = w(e) 1. The e-cotracted digraph D is slightly more complicated. For the vertex set of D we replace the two vertices u ad v by a sigle vertex uv. Ay edge 11

12 a b a b w(e) c u v d w(e)-1 c u v d a uv c b d D D D Figure 1: Reductio ad cotractio of a o-loop edge e i a weighted digraph i D of the form (x, u) becomes a edge (x, uv) i D. Also, ay edge i D of the form (v, y) becomes a edge (uv, y) i D. All other edges icidet to either u or v (icludig loops) are deleted. If (v, u) happes to be a edge i D, it becomes a loop at uv i D. All other edges (ot icidet to u or v) i D are retaied i D. No edge weights of survivig edges are chaged by the cotractio operatio. Next, suppose e = (u, u) is a loop at u i D. The we do the followig. As before, i the e-reduced digraph D, the weight of e is reduced by 1. To form the e-cotracted digraph D, we simply delete the vertex u ad all edges icidet to u. Usig the represetatio of B D (x, y) i (13), we will ow show that drop polyomials obey a reductio/cotractio rule. Theorem 3. Let D be a weighted digraph ad let e be a edge of D. Deote the e-reduced ad e-cotracted digraphs by D ad D, respectively. (i) If e is a o-loop edge the B D (x, y) = B D (x, y) + B D (x + w(e) 1, y); (15) 12

13 a b a b a b u w(e) u w(e)-1 c d c d c d D D D Figure 2: Reductio ad cotractio of a loop edge e i a weighted digraph (ii) If e is a loop the B D (x, y) = B D (x, y) + yb D (x + w(e) 1, y). (16) Proof. Let w, w, w deote the edge weights i D, D, D, respectively. We first cosider the case that e = (u, v) is a o-loop edge. From the defiitio (13), we have 13

14 B D (x, y) = ( ) x + w(s) (1 y) S y c(s) ( ) x + w(s) (1 y) S y c(s) + e S ) (1 y) S y c(s) + e S S P P (V ) = e S ( ( x + w(s) = e S ( x + w (S) e S = B D (x, y) ( x + w (S) e S + ( x + w(s) e S ) (1 y) S y c(s) + e S ) (1 y) S y c(s) ( ) x + w(s) (1 y) S y c(s) ( ) x + w (S) )(1 y) S y c(s) ( x + w(s) ) (1 y) S y c(s) ) (1 y) S y c(s) sice for e S, we have w(s) = w (S) for ay partial permutatio S. We ow use the fact that if e S, we have w (S) = w(s) 1. Hece B D (x, y) = B D (x, y) ( ) x + w(s) 1 (1 y) S y c(s) e S + ( ) x + w(s) (1 y) S y c(s) e S = B D (x, y) + ( ) x + w(s) 1 (1 y) S y c(s) 1 e S = B D (x, y) + ( ) x + w(s ) + w(e) 1 (1 y) 1 S y c(s ) 1 S as desired. = B D (x, y) + B D (x + w(e) 1, y) Next, we cosider the case that e = (u, u) is a loop. The first part of the derivatio is very similar to the precedig case ad we wo t repeat it. For 14

15 the secod part, we have B D (x, y) = ( ) x + w(s) (1 y) S y c(s) + ( x + w(s) e S e S = B D (x, y) ( ) x + w (S) (1 y) S y c(s) e S + ( ) x + w(s) (1 y) S y c(s) S = B D (x, y) ( x + w (S) + w(e) 1 S S=e S ( ) x + w(s ) + w(e) (1 y) 1 S y c(s )+1 + S S=e S = B D (x, y) + y S ( x + w(s ) + w(e) 1 1 = B D (x, y) + yb D (x + w(e) 1, y). This completes the proof of Theorem 3. ) (1 y) S y c(s) ) (1 y) 1 S y c(s )+1 ) (1 y) 1 S y c(s ) 6 Relatig the drop polyomial to the cover polyomial As metioed i the itroductio, the cover polyomial C D (x, y) for a weighted digraph D = (V, E, w) is defied by (6). I the case that D is simple, that is, w(e) = 1 for every e E, the (6) reduces to (5): C D (x, y) = i,j c D (i, j)x i y j, where we recall that c D (i, j) couts the umber of path/cycle covers of D cosistig of i paths ad j cycles. It was show i [5] that for e E, if D \ e deotes the digraph obtaied from the simple digraph D by deletig the edge e, ad D/e deotes the 15

16 correspodig e-cotracted digraph the we have the deletio/cotractio recurreces: (i) If e is a o-loop edge the C D (x, y) = C D\e (x, y) + C D/e (x, y); (ii) If e is a loop the C D (x, y) = C D\e (x, y) + yc D/e (x, y). Also, it is easy to see that C I() = x where I() deotes the -vertex digraph havig o edges. However, these are the same recurreces that B D (x, y) satisfies for simple digraphs D, sice every edge e E has w(e) = 1, so that whe we reduce its weight by 1, it becomes a weight 0 edge, which we kow by Theorem 1 we ca remove. Thus, we have Theorem 4. For simple digraphs D = (V, E), we have B D (x, y) = C D (x, y). Proof. The proof precedes by iductio o the umber of edges usig the reductio/cotractio rule i Theorem 3. We have see that for the base case D = I() with o edges: B I() (x, y) = x = C I() (x, y). Sice B D (x, y) ad C D (x, y) satisfy the same recurreces for reductio ad cotractio, the they are equal for all simple digraphs D. For digraphs with arbitrary edge weights, the deletio/cotractio rules for C D are the followig, where D \ e ad D/e are defied above (see [6]): (i) If e is a o-loop edge the C D (x, y) = C D\e (x, y) + w(e)c D/e (x, y); 16

17 (ii) If e is a loop the C D (x, y) = C D\e (x, y) + y w(e)c D/e (x, y). Thus, the deletio/cotractio recurreces for C D (x, y) ad the reductio/cotractio recurreces for B D (x, y) are quite differet for geeral weighted digraphs. However we have see i Theorem 4 that B D (x, y) ad C D (x, y) are equal whe D is a simple digraph. However, this is ot the oly time they ca be equal. We show a example i Figure 1 for a geeral weighted digraph D havig 2 vertices. A little computatio shows: B D (x, y) = x 2 + ((α + β)y + γ + δ 1)x + αβy 2 + ((1/2)α 2 + (1/2)β 2 + γδ (1/2)α (1/2)β)y + (1/2)γ 2 + (1/2)δ 2 (1/2)γ (1/2)δ, C D (x, y) = x 2 + ((α + β)y + γ + δ 1)x + αβy 2 + γδy. Takig the differece we fid B D (x, y) C D (x, y) = (1/2)((α(α 1) + β(β 1))y + γ(γ 1) + δ(δ 1)). Certaily this differece is 0 if each of α, β, γ ad δ is either 0 or 1. However, it is also 0 whe α = 2/5, β = 6/5, γ = 1/5, δ = 3/5, for example. α γ δ β α δ γ β D M Figure 3: A small geeral weighted digraph D ad its associated matrix M. 7 A reciprocity theorem The goal here is to prove a reciprocity theorem for weighted digraphs. Let J() deote the -by- matrix of all 1 s. 17

18 Theorem 5. For a weighted digraph D o vertices, we have B J() D (x, y) = ( 1) B D ( x y, y). (17) Proof. It follows from the alterative defiitio of B D (x, y) i Theorem 2 that it is a polyomial i x, y ad the weights w(e), e E. Thus, to prove the reciprocity theorem i (17) for geeral digraphs, it suffices to prove it for digraphs havig o-egative iteger weights. This will the imply that (17) holds for arbitrary weights. We will proceed by iductio o the umber of edges i D. The base case for I() holds sice the reciprocity theorem holds for the cover polyomial C D (x, y) for simple digraphs (see [6] for a detailed proof) ad C D (x, y) = B D (x, y) for simple digraphs D as show i Theorem 4. Suppose e = (u, v) is a edge i D. We first cosider the case that e is a o-loop edge. We apply the reductio/cotractio rule with D ad D as give i Theorem 3. B D (x, y) = B D (x, y) + B D (x + w(e) 1, y) = ( 1) B J() D ( x y, y) + ( 1) 1 B J( 1) D ( x y + w(e) 1, y). We ow apply the reductio/cotractio rule (15) for J() D usig the edge e. Note that the e-reductio of J() D is just J() D. Furthermore, the e-cotractio of J() D is J( 1) D. Therefore, B D (x, y) = ( 1) B J() D ( x y, y) + ( 1) 1 B J( 1) D ( x y + w(e) 1, y) ) = ( 1) (B J() D ( x y, y) + B J( 1) D ( x y + w(e) 1, y) + ( 1) 1 B J( 1) D ( x y + w(e) 1, y) = ( 1) B J() D ( x y, y) as desired. For the case that e is a loop, the proof is quite similar ad is omitted. A immediate cosequece of (17) is the followig iterestig idetity: 18

19 Corollary 3. B J() (x, y) = S P P (v) 8 Cocludig remarks ( ) x + S (1 y) S y cyc(s) = (x + y), There are may questios that are suggested by the precedig results. For example, although B D (x, y) ad C D (x, y) are quite differet whe evaluated o geeral weighted digraphs, they agree whe D is simple. Ca those D for which B D (x, y) = C D (x, y) be characterized? The fact that B D (x, y) ad C D (x, y) both satisfy the same reciprocity formula for geeral D (see (7)) suggests that somethig more fudametal is resposible for this behavior. It would be of iterest to uderstad the computatioal complexity of evaluatig B D (x, y). For example, for simple D, the coefficiet of x 0 y 1 term is just the umber of Hamiltoia cycles i D, somethig which is kow to be #P-hard to compute. (This follows from the fact that B D (x, y) = C D (x, y) for simple digraphs D.) How hard is it to evaluate B D (x, y) at specific poits i the (x, y) plae? For example,i the case that D is simple, B D (0, 0) = 0. However, for geeral weighted D, eve the value of B D (0, 0) does ot seem to be easy to compute. For the case of the Tutte polyomial, we metio the followig result of Jaeger, Vertiga ad Welsh [10]: Theorem 6. The problem of evaluatig the Tutte polyomial T G (x, y) of a geeral graph G at a poit (a, b) is #P -hard except whe (a, b) is o the special hyperbola (x 1)(y 1) = 1 or whe (a, b) is oe of the eight special poits (1, 1), ( 1, 1), (0, 1), ( 1, 0), (i, i), ( i, i), (j, j 2 ) ad (j 2, j), where j = e 2πi 3. I each of these exceptioal cases, the evaluatio ca be doe i polyomial time. I the case of the cover polyomial C D (x, y) (see [5]), the situatio is eve worse! I this case, Bläser ad H. Dell [1] have show that it is #P-hard 19

20 to evaluate C D (x, y) for geeral digraphs (with multiple edges ad loops) at a poit (a, b) uless (a, b) is oe of the three special poits (0, 0), (0, 1) ad (1, 1) (ad for these three poits, the computatio ca be doe i polyomial time). Presumably, the same result also may hold whe restricted to simple digraphs D. We poit that the results for weighted digraphs preseted i this paper ca be restated etirely i terms of matrices with etries i some commutative rig R with idetity. For if D = (V, E, w) is our digraph o vertices, the as we have metioed earlier we ca defie a matrix M = M(D) which is idexed by the elemets of V. For u, v V, the (u, v) etry of M is M(u, v) = w(e) if e = (u, v) E, ad is 0 if (u, v) / E. I terms of matrix operatios o the matrix M = M(D), we ca defie the reductio ad cotractio rules i the same way as those for weighted digraphs. The the same reciprocity theorems for digraphs (i.e., (3) ad (17)) apply to the correspodig matrix drop polyomial B M (x, y) as well. I aother directio, it would be quite iterestig to exted may of the kow results for permutatio statistics (e.g., see [13] or [11]) to the case of partial permutatios. Some results i this directio have recetly appeared i [4], for example. However, eve the most basic questios cocerig partial permutatios remai uaswered at preset. For example, suppose we let N (s, c) deote the umber of partial permutatios o [] which have size s ad cotai c cycles. That is, N (s, c) = {S [] : S is a partial permutatio, S = s, cyc(s) = c}. Whe s = the N (, c) is just equal to the familiar (usiged) Stirlig umber [ c] (see [9]). For example, [ 1] = ( 1)! ad [ 2] = ( 1)!( ). However, for 0 < s <, it is ot hard to show that N 1 (s, 0) = ( s) ( 1) s ad N (s, 1) = ( ) s ( 1) s ( ). It is possible s s+1 1 to derive explicit expressios for N (s, 2) ad N (s, 3) but the formulas seem to be gettig icreasigly upleasat! Is there ice way of expressig N (s, k) i geeral? 20

21 Of course, sice there are just ( s) s possible partial permutatios o [] of size s, the 0 c N (s, c) = ( ) s. s However, the followig idetity (equivalet to Corollary 2) seems less obvious. N (s, c)(1 y) s y c =!. s,c If the vertex set V of D is totally ordered, say V = [], we ca defie exc(s), the umber of exceedeces of a partial permutatio S as follows. Namely, defie exc(s) to be the umber of idices i such that σ(i) > i for the associated mappig σ : [] []. From the matrix perspective, this is the umber of etries of S which lie above the diagoal of M. How may partial permutatios S of size k have exc(s) = d? For the usual case of permutatios of [], these umbers are the usual Euleria umbers (see [9]). Of course, the same questios ca be asked whe V is just partially ordered, or more geerally, whe D is a arbitrary simple digraph. There is clearly much more to do. Refereces [1] M. Bläser ad H. Dell, Complexity of the Cover Polyomial, Automata, Laguages ad Programmig, Lecture Notes i Computer Sciece 4596, (2007), [2] J. P. Buhler ad R. L. Graham, A ote o the drop polyomial of a poset, Jour. Combiatorial Th. (A) 66 (1994), [3] J. P. Buhler ad R. L. Graham, Jugglig patters, passig ad posets, Mathematical Advetures for Studets ad Amateurs, MAA (2004),

22 [4] A. Claesso, V. Jelíek, E. Jelíková ad S. Kitaev, Patter avoidace i partial partitios, Electroic Jour. of Combiatorics, 18, o. 2, (2011), #P5. [5] F. R. K. Chug ad R. L. Graham, O the Cover Polyomial of a Digraph, Jour. Combiatorial Th. (B) 65 (1995), [6] F. R. K. Chug ad R. L. Graham, The matrix cover polyomial, Joural of Combiatorics 7 (2016), [7] F. R. K. Chug ad R. L. Graham, The biomial drop polyomial of a digraph, (preprit), fa/wp/diw 2.pdf. [8] O. M. D Atoa ad E. Muarii, The cycle-path idicator polyomial of a digraph, Advaces i Applied Math. 25 (2000), [9] R. L. Graham, D. E. Kuth ad O. Patashik, Cocrete Mathematics, A Foudatio for Computer Sciece, Addiso-Wesley, Bosto, 1994, xiii+657 pp. [10] F. Jaeger, D. L. Vertiga ad D. J. A. Welsh, O the computatioal complexity of the Joes ad Tutte polyomials, Math. Proc. Cambridge Philos. Soc. 108, (1990), [11] A. Medes ad J. Remmel, Coutig with Symmetric Fuctios, Spriger, Heidelberg, 2015, x+292 pp. [12] R. P. Staley, Acyclic orietatios of graphs, Discrete Math. 5 (1973), [13] R. P. Staley, Eumerative Combiatorics. Vol 1, secod editio, Cambridge Studies i Advaced Mathematics, 62 Cambridge Uiv. Press, Cambridge, 2012, xiv+626 pp. 22

23 [14] D. J. A. Welsh, The Tutte polyomial, Radom Structures ad Algorithms, 15, (1999),