Approximate formulation of the probability that the Determinant or Permanent of a matrix undergoes the least change under perturbation of a single element

Approximate formulation of t he pr obabi lity that the Deter minant or Permanent of a m atri x underg o es the l east change under p ert urbati on of a sing le el ement Gen ta Ito ∗ Maruo L ab., 500 El Camino R e al #302, Burlingame, CA 94010, United States. SUMMAR Y In an earlier pap er, w e discussed t h e probability that the determinant of a matrix undergo es the least change up on p erturbation of one of its elemen ts, provided that most or all of the elements of the matrix are c hosen at random and that the randomly c hosen elemen ts hav e a fixed probabilit y of b eing non-zero. I n this pap er, w e derive approximate form ulas for that probabilit y by assuming that the terms in the p ermanent of a matrix are indep end ent of one another, and we apply that assumption to sev eral classes of matrices. In th e course of deriving those form ulas, we identified several integer sequences t hat a re not listed on Sloane’s W eb site. Preprint submitted to Numer. Linear Algebra Appl. on June 28, 2007 Pr ep ar e d using nlaauth.cls key wo rds: Determinant; Permanen t; Perm utation; Sloane’s sequ ences 1. Intro duction In an earlier pap er [1], we discussed the problem of finding the probability t hat the determinant of a ma trix undergo es the least change under per turbation of one of its elements. In this pa p e r, we consider only the case where the randomly chosen matr ix elemen ts are v alues o f a contin uous (real) random v ariable, and we derive appr oximate fo r mulas for that pr obability by assuming that the terms in the p erma nent of a matrix are mutually indep endent. Denote the determina nt of any matrix A b y det A . F o r an ( n + 1) × ( n + 1) matr ix M n +1 , the ex pa nsion of de t M n +1 via row i is det M n +1 = m i 1 M i 1 + · · · + m ij M ij + · · · + m in +1 M in +1 , (1) where m ij is the element of M n +1 at the intersection of r ow i a nd column j , and M ij is the cofactor of m ij . M ij can b e written as ( − 1) i + j det S n , where S n is the n × n submatrix o f M n +1 which is obtained by deleting row i and co lumn j . Thus the probability that det M n +1 undergo es the lea s t change up on p erturba tion of element m ij is equal to the pr obability that ∗ Corresp ondence to: M aruo Lab., 500 El Camino R eal #302, Burlingame, CA 94010, United States. (Email: gito@maruolab.com) Preprint submitted to N umer. Linear Algebra Appl. on June 28, 2007 Pr ep ar e d using nlaauth.cls 2 G. ITO | det S n | is as small as p ossible. As in the earlier pap er, we treat the following thr e e cla sses of n × n matrices S n : (i) Matrices A n in which all the elements are v a lues of m utually independent random v aria ble s each o f which has probability r of be ing non-zer o and probability 1 − r of being 0 , where 0 < r < 1 . (ii) Matrices B n in which all but one o f the diag onal elements ar e set to 1 (i.e., b ii = 1 for i 6 = 1, where b 11 is the sp ecia l diagonal element), and b 11 and all the off-diago nal elements are as in (i). Thus B n =        b 11 b 12 b 13 · · · b 1 n b 21 1 b 23 · · · b 2 n b 31 b 32 1 · · · b 3 n . . . . . . . . . . . . . . . b n 1 b n 2 b n 3 · · · 1        (2) (iii) Matrices C n in which all the diag o nal elements are s et to 1 a nd all the off-dia gonal elements are as in (i). Thus C n =        1 c 12 c 13 · · · c 1 n c 21 1 c 23 · · · c 2 n c 31 c 32 1 · · · c 3 n . . . . . . . . . . . . . . . c n 1 c n 2 c n 3 · · · 1        The matrice s S n describ ed in (i), (ii), and (iii) ab ove will be said to be matrices of t yp e A n , B n , and C n , resp ectively . Every randomly chosen element of a matrix of any of these types (i.e., every element of a matrix of t yp e A n , e very off-diago na l element of a matr ix of type B n or C n , and the sp ecial diagona l element b 11 of a matr ix of type B n ) will b e called a variable e le ment ; the rema ining elements will b e called fi xe d e lements. If X is a contin uous set such that 0 ∈ X (as is assumed here ), and X is a ra ndom v ar iable whose v alues ar e selected fro m X , then the probability function P for X is defined by P ( X = 0) = 1 − r P ( X 6 = 0) = r (3) Since X is contin uous, P ca n be expressed in terms of a density function q as D ⊆ X ⇒ P ( X ∈ D ) = Z D q ( x ) dx , (4) where 0 ≦ q ( x ) ≦ 1 , Z { 0 } q ( x ) dx = 1 − r , Z X −{ 0 } q ( x ) dx = r If q is contin uous on X − { 0 } (a s is assumed here), then P ( X = x ) =  1 − r , x = 0 0 , x ∈ X − { 0 } (5) 2 APPRO X IMA TE FORMULA TION OF THE PROBABILITY 3 As found in [1], the probability that det M n +1 undergo es the least change up o n p ertur bation of e lement m ij is equal to the probability that det S n = u , where u = 0 if S n is of type A n or B n , and u = 1 if S n is of t yp e C n . In [1], the pr obability that det S n = u was fo r mulated in terms o f binary matr ic es (matrices of 0 ’s and 1’s ), as follows: F or a matrix S n of type A n , B n , or C n , ˜ S n is the n × n binary matrix with ˜ s ij = 1 if s ij 6 = 0, and ˜ s ij = 0 if s ij = 0. Moreov er, ˜ s ij is a variable (resp. fixe d ) element of ˜ S n if s ij is a v ariable (resp. fixed) element o f S n , and ˜ S n is o f type ˜ A n (resp. ˜ B n , ˜ C n ) if S n is o f type A n (resp. B n , C n ). Every v aria ble element of ˜ S n has pro ba bility r o f b eing 1, and probability 1 − r of b eing 0. Deno te the p ermanent of any matrix A by p er A . The expansion of the p erma nent of ˜ S n via the per mutation g roup S n on the set { 1 , 2 , 3 , . . . , n } is per ˜ S n = X σ ∈ S n ˜ s σ (1)1 ˜ s σ (2)2 · · · ˜ s σ ( n ) n (6) Since ˜ S n is a binar y matrix, every ter m in this expansion is either 0 or 1, so the v alue of p er ˜ S n is a non-neg ative integer. In [1], it was sho wn that the probability that det S n = u is equal to the probability that p er ˜ S n = u , and that ˜ S n has non-ze r o (p ositive) pro bability of having per manent u if and only if every term in the expansio n o f p er ˜ S n via S n that contains at leas t one v aria ble element is 0. Thus if S n is of t yp e A n or B n , then S n has non-zer o probability of having determinant 0 if and only if every term in the expansion of p er ˜ S n is 0; and if S n is of t yp e C n , then S n has non-zer o probability of having determinant 1 if and o nly if the only non- zero term in the expansio n of p er ˜ S n is the “diago na l” ter m, ˜ c 11 ˜ c 22 ˜ c 33 · · · ˜ c nn . A matrix of t yp e ˜ A n can b e conv e r ted to a matrix of type ˜ B n by repla cing all but o ne of the dia gonal elements with a fixed element 1. The latter matrix can in tur n b e converted to a matrix of type ˜ C n by replacing the sole v ariable elemen t on the main diagonal (element ˜ b 11 ) with a fixed element 1. Therefore, we migh t exp ect to find that there exist analytic formulas for P p er ˜ B n =0 ( r ) in terms of P p er ˜ A n =0 ( r ), and P p er ˜ C n =1 ( r ) in terms of P p er ˜ B n =0 ( r ), where P p er ˜ S n = u ( r ) denotes the proba bility that per ˜ S n = u . In [1], the exact probability P p er ˜ S n = u ( r ) fo r a sp ecific t yp e of matrix ( ˜ A n , ˜ B n , or ˜ C n ) was formulated in terms of the num b er s of matrices ˜ S n of tha t type which have i v aria ble elements with a v alue o f 1 in the expansio n of their p ermanent (where i ranged from 0 to some i max that dep ended on the type of matrix). In this pap er , we will derive an approximate proba bility Q p er ˜ S n = u ( r ) that p er ˜ S n = u , by assuming tha t all the terms in the expansion of per ˜ S n via S n are indep endent of one another . F o r example, the expansion of the per manent of a ma trix of t yp e ˜ A 3 is per ˜ A 3 = ˜ a 11 ˜ a 22 ˜ c 33 + ˜ a 13 ˜ a 32 ˜ a 21 + ˜ a 12 ˜ a 23 ˜ a 31 + ˜ a 13 ˜ a 31 ˜ a 22 +˜ a 11 ˜ a 23 ˜ a 32 + ˜ a 12 ˜ a 21 ˜ a 33 If ˜ a 23 = 0, the terms ˜ a 12 ˜ a 23 ˜ a 31 and ˜ a 11 ˜ a 23 ˜ a 32 m ust b oth b e zero, but we will ignore that relationship. Ins tead, for every m ≤ n and ea ch of the three types of matrices ( ˜ A n , ˜ B n , and ˜ C n ), we will cons ider o nly the num b er of terms in the expans ion o f the p er manent of a matrix of that type which have m v ariable elements, regar dless of the v alues of those v ariable elements or the connections b etw een differen t terms in the ex pansion. 3 4 G. ITO 2. F ormulation of Q p er ˜ S n = u ( r ) There a re n ! terms ˜ s σ (1)1 ˜ s σ (2)2 · · · ˜ s σ ( n ) n in (6 ), beca use the num b er o f elements σ ∈ S n is n !. F or each term, let m b e the n umber of v ar iable elemen ts it con tains, and let E n ( m ) be the nu mber of terms with m v ariable elements. Then Q p er ˜ S n = u ( r ) = n Y m =1 (1 − r m ) E n ( m ) , where r m is the proba bility that a ll the v aria ble e lements in a term with m v a r iable ele ment s are non-zero, 1 − r m is the pro bability that at least one v ariable elemen t in a term with m v aria ble e le ment s is 0, and (1 − r m ) E n ( m ) is the proba bilit y that a ll the terms with m v ar iable elements are 0. What remains to b e done is to determine E n ( m ) for every m ≤ n and each of the three types of matrices. 2.1. T yp e ˜ A n F or type ˜ A n , every term in the expa nsion of per ˜ A n has n v ariable element s, so E n ( m ) = 0 for every m < n . Since there are n ! terms, w e obtain Q p er ˜ A n =0 ( r ) = (1 − r n ) n ! (7) 2.2. T yp e ˜ C n F or type ˜ C n , let W n ( m ) denote E n ( m ), so that Q p er ˜ C n =1 ( r ) = n Y m =1 (1 − r m ) W n ( m ) (8) Prop ositi on 2.1. W n ( n − 1) = n · W n − 1 ( n − 1) (9) Pro of Ev ery ter m in the expansion of p er ˜ C n that ha s n − 1 v aria ble elements contains just one fixed (diag onal) element (namely , ˜ c ii for some i with 1 ≤ i ≤ n ). F or every i , there exists Figure 1. Cross shapes deleted from ˜ C n to create the ( n − 1) × ( n − 1) submatrices ˜ C i n of ˜ C n a na tural one-to-one corresp ondence betw een the set T 1 i of terms in the expans ion of p er ˜ C n that con tain elemen t ˜ c ii (and ha ve n − 1 v ariable elemen ts of ˜ C n apiece) and the set T 2 i of terms in the ex pansion of pe r ˜ C i n that hav e n − 1 v a riable ele ments each, where ˜ C i n is the ( n − 1 ) × ( n − 1) submatrix of ˜ C n which is formed by deleting the “ cross shap e” that consists 4 APPRO X IMA TE FORMULA TION OF THE PROBABILITY 5 of row i a nd column i . There are n s uch c r oss shap es (one for each dia gonal element of ˜ C n ), as shown in Fig. 1; moreover, for every i, til deC i n is a matrix of type ˜ C n − 1 . Thus we obtain (9).  Prop ositi on 2.2. F or every m with 1 ≦ m ≦ n − 1 , W n ( m ) = n P m m ! · W m ( m ) (10) Pro of Let m, k b e such that 1 ≤ m ≤ n − 1 and k = n − m . A term in the expansion of per ˜ C n has m v a riable elemen ts if and only if it has k fixed (diagonal) elemen ts (namely , ˜ c i 1 i 1 , . . . , ˜ c i k i k for some i 1 , . . . , i k with 1 ≤ i 1 < i 2 < · · · < i k ≤ n ). F or every k -tuple ( i 1 , . . . , i k ) with 1 ≤ i 1 < i 2 < · · · < i k ≤ n , ther e exists a natura l one-to-o ne cor resp ondence b etw een the set T 1 , { i 1 ,...,i k } of terms in the expansion of p er ˜ C n that contain element s ˜ c i 1 i 1 , . . . , ˜ c i k i k (and hav e m v ariable elemen ts of ˜ C n apiece) and the s et T 2 , { i 1 ,...,i k } of terms in the expansion of per ˜ C { i 1 ,...,i k } n that have m v a riable e le ment s eac h, where ˜ C { i 1 ,...,i k } n is the m × m submatrix of ˜ C n which is formed b y deleting the cross sha pe s that corresp o nd to i 1 , . . . , i k (i.e., the submatrix of ˜ C n which is formed by deleting rows i 1 , . . . , i k and co lumns i 1 , . . . , i k ). The nu mber of k -tuples ( i 1 , . . . , i k ) with 1 ≤ i 1 < i 2 < · · · < i k ≤ n is n C k ; mor e over, for each such k -tuple, til deC { i 1 ,...,i k } n is a matrix of type ˜ C m . Also, n C k = n P m m ! , since k = n − m and n C n − m = n ! ( n − m )! m ! = n P m m ! . Thus we obtain (10 ).  Prop ositi on 2.3. F or n ≥ 1 , W n ( n ) = n − 1 X j =1 ( − 1) n − j +1 · n P j − 1 (11) Pro of W n ( n ) is the num b er of p ermutations of the num b ers 1 , 2 , 3 , . . . , n with no fixed p o int s (i.e., the n umber of p er mutations that hav e no cycles of length one). Such p ermutations are called der angements . In 1713, Pierre de Mont mort [2] pr ov ed that W n ( n ) = n · W n − 1 ( n − 1 ) + ( − 1) n (12) 5 6 G. ITO Thu s we obtain (1 1) recursively as follows: W n ( n ) = n · W n − 1 ( n − 1) + ( − 1) n (13) = n P 1 · W n − 1 ( n − 1 ) + ( − 1) n · n P 0 = n ·  ( n − 1 ) W n − 2 ( n − 2) + ( − 1) n − 1  + ( − 1) n = n ( n − 1 ) · W n − 2 ( n − 2 ) + n · ( − 1) n − 1 + ( − 1) n = n P 2 · W n − 2 ( n − 2 ) + 2 X j =1 ( − 1) n − j +1 · n P j − 1 = · · · = n P k · W n − k ( n − k ) + k X j =1 ( − 1) n − j +1 · n P j − 1 = · · · = n P n − 1 · W 1 (1) + n − 1 X j =1 ( − 1) n − j +1 · n P j − 1 = n − 1 X j =1 ( − 1) n − j +1 · n P j − 1 The last step follows from the fact tha t W 1 (1) = 0.  Prop ositi on 2.4. F or every m with 0 ≤ m ≤ n , W n ( m ) = n P m · m X l =0 ( − 1) l l ! (14) Pro of The only term in the expansion of p er ˜ C n that ha s no v a riable elements is ˜ c 11 · · · ˜ c nn , so W n (0) = 1 = n P 0 · ( − 1) 0 0! . This prov es that (14) ho lds for m = 0. By (1 1), W n ( n ) = n − 1 X j =1 ( − 1) n − j +1 · n P j − 1 = n − 1 X j =1 ( − 1) n − j +1 · n ! ( n − j + 1 )! = n ! · n − 1 X j =1 ( − 1) n − j +1 ( n − j + 1 )! = n P n · n X l =2 ( − 1) l l ! = n P n · n X l =0 ( − 1) l l ! 6 APPRO X IMA TE FORMULA TION OF THE PROBABILITY 7 This proves that (14) holds for m = n (t he last step follows from the fact that 1 X l =0 ( − 1) l l ! = 1 − 1 = 0), so we can as s ume tha t 1 ≤ m ≤ n − 1. By (10) and (11 ), W n ( m ) = n P m m ! · W m ( m ) = h n ! ( n − m )! i m ! · m − 1 X j =1 ( − 1) m − j +1 · m P j − 1 = h n ! ( n − m )! i m ! · m − 1 X j =1 ( − 1) m − j +1 · m ! ( m − j + 1 )! = n ! ( n − m )! · m − 1 X j =1 ( − 1) m − j +1 ( m − j + 1 )! = n P m · m X l =2 ( − 1) l l ! = n P m · m X l =0 ( − 1) l l !  The v a lues of W n ( m ) fo r n = 1 , . . . , 6 (and m = 0 , . . . , n ) ar e given in T able I. The T able I . V alues of W n ( m ) for n = 1 , . . . , 6 ( an d m = 0 , . . . , n ) n \ m 0 1 2 3 4 5 6 1 1 − 1 − → 0 ↓ × 2 2 ↓ × 2 2 1 0 +1 − → 1 ↓ × 3 3 ↓ × 3 2 ↓ × 3 3 1 0 3 − 1 − → 2 ↓ × 4 4 ↓ × 4 3 ↓ × 4 2 ↓ × 4 4 1 0 6 8 +1 − → 9 ↓ × 5 5 ↓ × 5 4 ↓ × 5 3 ↓ × 5 2 ↓ × 5 5 1 0 10 20 45 − 1 − → 44 ↓ × 6 5 ↓ × 6 5 ↓ × 6 4 ↓ × 6 3 ↓ × 6 2 ↓ × 6 6 1 0 15 40 135 264 +1 − → 265 ↓ × 7 7 ↓ × 7 6 ↓ × 7 5 ↓ × 7 4 ↓ × 7 3 ↓ × 7 2 ↓ × 7 sequence { W n ( n ) } n ≥ 0 is num b er A00 0166 in Sloane’s list [3]. The fir st ( n = 0) term, which has no meaning for o ur purp os es, is given there as 1. F or n ≥ 1, W n ( n ) is not only the n umber o f dera ngements o f the num b ers 1 , 2 , 3 , . . . , n , but a lso the p ermanent of the n × n binary matrix with 0’s on the main diagonal and 1 ’s ev erywher e else. The sequences { W n (2) } n ≥ 2 , { W n (3) } n ≥ 3 , { W n (4) } n ≥ 4 , and { W n (5) } n ≥ 5 are given in Sloane’s list (num b ers A00217 , A0072 90, A060 0 08, and A06083 6, re sp ectively). There is no m > 5 for which the sequence { W n ( m ) } n ≥ m is given in that list. An alter native metho d of determining the v alue of W n ( m ) is to use the cycle structure of the p er mutation group S n . F or n = 5, t he types and num b ers of cy c le s asso ciated with elements of S 5 that are applica ble to each v alue of m are listed in T able I I, together with the v alues of W 5 ( m ) computed from them. F or m = 4 and m = 5, there are tw o different t yp es of cycles each. The p ermutations tha t corre s p ond to 7 8 G. ITO m = 4 (which index the ter ms in the expans ion of p er ˜ S n that hav e 4 v ar iable e lements) ar e those tha t can b e expressed as a 4 -cycle (such as (1234)) and those that can b e expressed as a pro duct of tw o 2-cycles (such as (12)(34)). Similarly , the permutations that corresp ond to m = 5 (whic h index the terms with 5 v ariable elemen ts) are the o ne that ca n b e expressed as the 5-cy cle (123 45) and those t hat can b e expressed as a product of o ne 2 -cycle and one 3-cycle (such as (12)(34 5)). T able I I. Computation of W 5 ( m ) from group table for S 5 m cycle rep. No. of cycles W 5 ( m ) 0 1 5 e 5! / ` 1 5 · 5! ´ = 1 1 2 2 1 · 1 3 (12) 5! / ` 2 1 · 1! · 1 3 · 3! ´ = 10 10 3 3 1 · 1 2 (123) 5! / ` 3 1 · 1! · 1 2 · 2! ´ = 20 20 4 4 1 · 1 1 (1234) 5! / ` 4 1 · 1! · 1 1 · 1! ´ = 30 30 + 15 2 2 · 1 1 (12)(3 4) 5! / ` 2 2 · 2! · 1! · 1 1 ´ = 15 = 45 5 5 1 (12345) 5! / ` 5 1 · 1! ´ = 24 24 + 20 2 1 · 3 1 (12)(3 45) 5! / ` 2 1 · 1! · 3 1 · 1! ´ = 20 = 44 2.3. T yp e ˜ B n F or type ˜ B n , let V n ( m ) denote E n ( m ), so that Q p er ˜ B n =0 = n Y m =1 (1 − r m ) V n ( m ) (15) Prop ositi on 2.5. F or every m with 1 ≤ m ≤ n , V n ( m ) = n P m n · (" ( m + 1) · m X l =0 ( − 1) l l ! # − ( − 1) m m ! ) (16) Pro of Let ˜ b 11 be the sole v ar iable element on the main diago nal of a matrix of t yp e ˜ B n . If we delete the L-shap e tha t consists of r ow 1 and column 1 , w e obtain a matrix of type ˜ C n − 1 . Therefore, V n ( m ) − W n − 1 ( m − 1) = W n ( m ) − W n − 1 ( m ) (17) By (14), V n ( m ) = W n ( m ) − W n − 1 ( m ) + W n − 1 ( m − 1) = " n P m · m X l =0 ( − 1) l l ! # − " n − 1 P m · m X l =0 ( − 1) l l ! # + " n − 1 P m − 1 · m − 1 X l =0 ( − 1) l l ! # = n P m n · (" ( m + 1) · m X l =0 ( − 1) l l ! # − ( − 1) m m ! )  The v alues of V n ( m ) for n = 1 , . . . , 8 (a nd m = 0 , . . . , n ) are given in T able I I I. The sequence { V n } n ≥ 0 is num b er A00025 5 in Slo ane’s list [3]. The first ( n = 0) term, whic h has no meaning for our purp oses, is given there as 1. F or n ≥ 1 , V n ( n ) is the num b er of p ermutations of the 8 APPRO X IMA TE FORMULA TION OF THE PROBABILITY 9 T able I I I . V alues of V n ( m ) for n = 1 , . . . , 8 ( an d m = 0 , . . . , n ) n \ m 0 1 2 3 4 5 6 7 8 1 0 1 2 0 1 1 3 0 1 2 3 4 0 1 3 9 11 5 0 1 4 18 44 5 3 6 0 1 5 30 110 265 309 7 0 1 6 45 220 795 1854 2119 8 0 1 7 63 385 1855 6489 14 833 1 6687 nu mbers 1 , 2 , 3 , . . . , n + 1 such that there is no ele ment k which is mapp ed to k + 1 ; it is also the per manent of the n × n binary ma trix that has 0’s in a ll but one of the elements on the main diagonal, and 1’s everywhere e ls e. The only v alue of m for which the sequence { V n ( m ) } n ≥ m is given in Sloane’s list is 3 (wher e the sequence { V n (3) } n ≥ 3 is n umber A04 5943 ). 3. Co mparison of F ormulas F or n = 3, the formulas for the approximate pro ba bilities Q p er ˜ S 3 = u ( r ) are Q p er ˜ A 3 =0 ( r ) = ( 1 − r 3 ) 3! = (1 − r 3 ) 6 Q p er ˜ B 3 =0 ( r ) = 3 Y m =1 (1 − r m ) V 3 ( m ) = (1 − r 1 ) 1 (1 − r 2 ) 2 (1 − r 3 ) 3 Q p er ˜ C 3 =1 ( r ) = 3 Y m =1 (1 − r m ) W 3 ( m ) = (1 − r 1 ) 0 (1 − r 2 ) 3 (1 − r 3 ) 2 The formulas for the exac t pr obabilities P p er ˜ S 3 = u ( r ) derived in [1] are P p er ˜ A 3 =0 ( r ) = (1 − r ) 9 + 9 r (1 − r ) 8 + 36 r 2 (1 − r ) 7 + 78 r 3 (1 − r ) 6 +90 r 4 (1 − r ) 5 + 45 r 5 (1 − r ) 4 + 6 r 6 (1 − r ) 3 P p er ˜ B 3 =0 ( r ) = (1 − r ) 7 + 6 r (1 − r ) 6 + 13 r 2 (1 − r ) 5 + 10 r 3 (1 − r ) 4 +2 r 4 (1 − r ) 3 P p er ˜ C 3 =1 ( r ) = (1 − r ) 6 + 6 r (1 − r ) 5 + 12 r 2 (1 − r ) 4 + 6 r 3 (1 − r ) 3 All six o f the functions given ab ove (for n = 3) are gra phed v s . r in Fig. 2. Also, Fig. 3 shows the six curves for n = 5 vs . r . F or every n ≥ 1 and all three types of matrices ( ˜ A n , til de B n , and ˜ C n ), n X m =0 E n ( m ) = n ! (18) Thu s the differences in the v a lues of E n ( m ) for the different t yp es of matrices s tem entirely from differences in the distribution of n ! ov e r the individual v alue s of m . F or type ˜ A n , that 9 10 G. ITO Figure 2. Graphs of Q p er ˜ S 3 = u ( r ) and P p er ˜ S 3 = u ( r ) vs. r Figure 3. Graphs of Q p er ˜ S 5 = u ( r ) and P p er ˜ S 5 = u ( r ) vs. r distribution is trivial (namely , E n ( n ) = n !, and E n ( m ) = 0 for m < n ). In t he case of t yp e ˜ C n , we found an analytic formula for the distribution: W n ( m ) = n P m · m X l =0 ( − 1) l l ! , m = 0 , 1 , 2 , . . . , n As illustra ted earlier , the distribution for type ˜ C n can a lso b e de r ived fro m the cyc le structure of the per mutation group S n . Moreover, we hav e shown that the distribution for type ˜ B n can b e derived from that for type ˜ C n , by using the relation V n ( m ) − W n − 1 ( m − 1) = W n ( m ) − W n − 1 ( m ). As es ta blished in [1], there is an analytic formula for the co e fficients in P p er ˜ C n =1 ( r ), s ince they ca n b e derived from c hara c ter istics of acy clic digraphs with vertex set { 1 , 2 , 3 , . . . , n } . F or several v alues of n , the co efficients in P p er ˜ A n =0 ( r ) and P p er ˜ B n =0 ( r ) were also presen ted in that pa p er , but they were determined by c o mputer, using an a lgorithm that too k a binary matrix o f a given type as input and computed its p ermanent. It rema ins to b e seen whether t here are explicit formulas for the co efficient s in P p er ˜ A n =0 ( r ) a nd P p er ˜ B n =0 ( r ), and whether there is an analytic formula that relates one set of co efficients to the other. REFERENCES 10 APPRO X IMA TE FORMULA TION OF THE PROBABILITY 11 1. Ito, G., Least change in the Determinant or Pe rmanent of a matrix under p erturbation of a single element: con tinu ous and discrete cases, Numeric al Line ar Algebr a wit h Applic ations (submitted). 2. de Mon tmort, P .R. Essay D ’ analyse Sur L es J eux De Hasar d (3rd edn). AMS/Chelsea Pub. Co.: New Y or k, 1980; 131–138. 3. Sl oane, N.J.A. The O n-Line Encyclop e dia of Inte ger Se quenc es (published electronically at h ttp://www.research.at t.com/ g njas/sequences/) . 1996–2007. 11