COUNTING OCCURRENCES OF 132 IN AN EVEN PERMUTATION

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1 IJMMS 00:5, PII. S Hindawi Publishing Corp. COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION TOUFIK MANSOUR Received 9 April 003 We study the generating function for the number of even or odd) permutations on n letters containing exactly r 0 occurrences of a 13 pattern. It is shown that finding this function for a given r amounts to a routine check of all permutations in S r. 000 Mathematics Subject Classification: 05A05, 05A15, 05C Introduction. Let [n] ={1,,...,n} and let S n denote the set of all permutations of [n]. We will view permutations in S n as words with n distinct letters in [n]. A pattern is a permutation σ S k, and an occurrence of σ in a permutation π = π 1 π π n S n is a subsequence of π that is order equivalent to σ. For example, an occurrence of 13 is a subsequence π i π j π k 1 i<j<k n) ofπ such that π j <π i <π k.wedenote by τπ) the number of occurrences of τ in π, and we denote by s r σ n) the number of permutations π S n such that σπ)= r. In the last decade much attention has been paid to the problem of finding the numbers s r σ n) for a fixed r 0 and a given pattern τ see [1,, 3, 5, 6, 7, 10, 11, 1, 15, 16, 17, 18, 19]). Most of the authors consider only the case r = 0, thus studying permutations avoiding a given pattern. Only a few papers consider the case r>0, usually restricting themselves to patterns of length 3. Using two simple involutions reverse and complement) ons n, it is immediate that, with respect to being equidistributed, the six patterns of length three fall into the two classes {13,31} and {13,13,31,31}. Noonan [13] proved that s 1 13 n) = 3 n n. 1.1) n 3 Noonan and Zeilberger [1] suggested a general approach to the problem; they gave another proof of Noonan s result, and conjectured that s13 n) = 59n +117n+100 n, nn 1)n+5) n n 3 s13 1 n) =. n 3 The second conjecture was proved by Bóna in [6] and the first conjecture was proved by Fulmek [8]. Noonan and Zeilberger conjectured that s r σ n) is P-recursive in n for any r and τ. It was proved by Bóna [] forσ = 13. Mansour and Vainshtein [11] suggested a new approach to this problem in the case σ = 13, which allows one to get an explicit 1.)

2 1330 TOUFIK MANSOUR expression for s13 r n) for any given r. More precisely, they presented an algorithm that computes the generating function n 0 s13 r n)xn for any r 0. Let π be any permutation. The number of inversions of π is given by i π = {i,j) : π i >π j, i < j}. Thesignature of π is given by signπ) = 1) iπ.wesayπ is an even permutation resp., odd permutation) ifsignπ) = 1 resp., signπ) = 1). We denote by E n resp., O n ) the set of all even resp., odd) permutations in S n. Clearly, E n = O n =1/)n! for all n. We denote by eσ r n) resp., oσ r n)) the number of even resp., odd) permutations π E n resp., π O n ) such that σπ)= r. Apparently, for the first time the relation between even odd) permutations and pattern-avoidance problem was suggested by Simion and Schmidt in [16] forσ S 3.In particular, Simion and Schmidt [16] proved that e 0 13 n) = 1 n+1) o 0 13 n) = 1 n+1) n + 1 n n+1 n 1 n n+1 n 1 n 1 n 1 n 1 n 1 where = 0 and n is an even number. n 1)/ In this note, as a consequence of [1], we suggest a new approach to this problem in the case of even or odd) permutations where σ = 13, which allows one to get an explicit expression for e13 r n) for any given r. More precisely, we present an algorithm that computes the generating functions E r x) = n 0 e13 r n)xn and O r x) = n 0 o13 r n)xn for any r 0. To get the result for a given r, the algorithm performs certain routine checks for each element of the symmetric group S r. The algorithm has been implemented in C, and yields explicit results for 0 r 6. ), ), 1.3). Definitions and preliminary results. Recall the definitions kernel permutation, kernel cell decomposition, feasible cells, shapes, kernel shapes, and cells) and the notations s the size of the kernel, c the capacity of the kernel, and f the number of the feasible cells in the kernel cell decomposition) which are given in [1]. In this section we describe how the cell decomposition approach see [1]) can be determined by the generating function for the number of even permutations which contain the pattern 13 exactly r times. Let π be any permutation with a kernel permutation ρ, and assume that the feasible cells of the kernel cell decomposition associated with ρ are ordered linearly according to, C 1,C,...,C fρ) see [1, Lemma 3]). Let d j be the size of C j. For example, let π = with kernel permutation ρ = 13, then d 1 =, d = 1, d 3 = 0, and d = 1 see Figure.1). We denote by l j ρ) the number of the entries of ρ that lie to the north-west from C j or lie to the south-east from C j. For example, let ρ = 13, as on Figure.1, then l 1 ρ) = 3, l ρ) =, l 3 ρ) = 3, and l ρ) =. Clearly, l 1 ρ) = sρ) 1andl fρ) = sρ) for any nonempty kernel permutation ρ. Define signc) = 1) 1C) for any cell C.

3 COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION 1331 m C 3 l Figure.1. Kernel cell decomposition for π = S13). Lemma.1. For any permutation π with a kernel permutation ρ, signπ) = 1) 1 i j fρ) d id j + fρ) d j l j ρ)) signρ) fρ) sign C j)..1) Proof. To verify this formula, we count the number of occurrences of the pattern 1 in π. There are four possibilities for an occurrence of 1 in π. The first possibility is an occurrence in one of the cells C j, so in this case there are fρ) 1Cj ) occurrences. The second possibility is an occurrence in kerπ, so there are 1ρ) occurrences. The third possibility is an occurrence of two elements, of which the first belongs to kerπ and the second belongs to C i, so there are fρ) d jl j ρ) occurrences see [1, Lemmas and 5]). The fourth possibility is an occurrence of two elements, of which the first belongs to C i and the second belongs to C j where i<j see [1, Lemmas and 5]), so there are 1 i<j fρ)d i d j occurrences. Therefore, fρ) signπ) = 1) 1Cj ) 1) 1ρ) fρ) 1) d j l j ρ) 1) 1 i<j fρ) d id j,.) equivalently, signπ) = 1) 1 i j fρ) d id j + fρ) d j l j ρ)) signρ) fρ) signcj ). We say that the vector v = v 1,v,...,v n ) is a binary vector if v i {0,1} for all i, 1 i n. We denote the set of all binary vectors of length n by n. For any v n, we define v =v 1 + v + +v n. For example, ={0,0),0,1),1,0),1,1)} and 1,1,0,0,1) =3. For any kernel permutation ρ and for a =±1, we denote by X a ρ) resp., Y ρ a ) the set of all the binary vectors v fρ) such that 1) v +sρ) = a resp., 1) v = a). For any v fρ), we define 1 i<j fρ) z ρ v) = 1) v iv j + fρ) l j ρ)v j signρ)..3) Letting ρ be any kernel permutation and v = v 1,v,...,v fρ) ) fρ),wedenoteby Sρ;v) the set of all permutations of all sizes with kernel permutation ρ such that the

4 133 TOUFIK MANSOUR corresponding cells C j satisfy 1) d j = 1) v j ;insuchacontextv is called a length argument vector of ρ. By definitions, the following result holds immediately. Lemma.. For any kernel permutation ρ, Sρ) = Letting ρ be any kernel permutation and letting v fρ) Sρ;v)..) v = v 1,v,...,v fρ) ), u = u1,u,...,u fρ) ) fρ),.5) we denote by Sρ;v,u) the set of all permutations in Sρ;v) such that the corresponding cells C j satisfy signc j ) = 1 if and only if u j = 0; in such a context u is called a signature argument vector of ρ. By Lemma., the following result holds immediately. Lemma.3. For any kernel permutation ρ, Sρ) = v fρ) Sρ;v) = v fρ) u fρ) Sρ;v,u)..6) For any a,b {0,1} we define 1 Er x)+ 1) a E r x) ), if b = 0, H r a,b) = 1 Or x)+ 1) a O r x) ), if b = 1..7) By definitions, the following result holds immediately. Lemma.. Let a,b {0,1}. Then the generating function for all permutations π such that 13π) = r, 1) π = 1) a, and signπ) = 1) b is given by H r a,b), where π denotes the length of the permutation π. 3. Main theorem. The main result of this note can be formulated as follows. Denote by K the set of all kernel permutations, and by K t the set of all kernel shapes for permutations in S t. Letting ρ be any kernel permutation, for any a,b {0,1} and any r 1,...,r fρ), we define L ρ r 1,...,r fρ) a,b) = fρ) v X ρ 1) a u Y ρ 1) b zρ v) H rj vj,u j ). 3.1) Theorem 3.1. Let r 1. For any a,b {0,1}, H r a,b) = ρ K r +1 r 1 + +r fρ) =r cρ), r j 0 L ρ r 1,...,r fρ) a,b). 3.) Proof. We fix a kernel permutation ρ K r +1, a length argument vector v = v 1,..., v fρ) ) X 1) aρ), and a signature argument vector u = u 1,...,u fρ) ) Y fρ) 1) b z ρv).

5 COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION 1333 Recall that the kernel ρ of any π contains exactly cρ) occurrences of 13. The remaining r cρ) occurrences of 13 are distributed among the feasible cells of the kernel cell decomposition of π. By[1, Theorem ], each occurrence of 13 belongs entirely to one feasible cell, and the occurrences of 13 in different cells do not influence one another. Let π be any permutation such that 13π) = r,signπ) = 1) b, and 1) π = 1) a, together with a kernel permutation ρ, length argument vector v, and signature argument vector u. ThenbyLemma.3, the cells C j satisfy the following conditions: 1) v j = 0 if and only if d j is an even number, ) u j = 0 if and only if signc j ) = 1, 3) 1) v 1+ +v fρ) +sρ) = 1) a, ) 1) u 1+ +u fρ) z ρ v) = 1) b. Therefore, by Lemma., this contribution gives x sρ) fρ) r 1 + +r fρ) =r cρ), r j 0 H rj vj,u j ). 3.3) Hence by Lemma.3 and [1, Theorem 1], summing over all the kernel permutations ρ K r +1, length argument vectors v X 1) aρ), and signature argument vectors u Y fρ), then we get the desired result. 1) b z ρv) Theorem 3.1 provides a finite algorithm for finding E r x) and O r x) for any given r 0, since we only have to consider all permutations in S r +1 and to perform certain routine operations with all shapes found so far. Moreover, the amount of search can be decreased substantially due to the following theorem. Theorem 3.. The only kernel permutation of capacity r 1 and size r +1 is ρ = r 1,r +1,r 3,r,...,r j 3,r j,...,1,,. 3.) Its parameters are given by sρ) = r + 1, cρ) = r, fρ) = r +, signρ) = 1, and z ρ v 1,...,v r + ) = 1) 1+v r ++ 1 i<j r + v i v j ). Proof. The first part of the theorem holds by [1, Proposition]. Besides, by using the form of ρ we get sρ) = r + 1, cρ) = r, fρ) = r +, signρ) = 1, and l j ρ) = r for all j = 1,,...,r + 1andl r + ρ) = r + 1. Therefore, z ρ v 1,...,v r + ) = 1) 1+v r ++ 1 i<j r + v i v j ). By this theorem, it suffices to search only permutations in S r. Below we present several explicit calculations The case r = 0. Westartfromthecaser = 0. Observe that Theorem 3.1 remains valid for r = 0, provided that the left-hand side of 3.) fora = b = 0isreplaced by H r 0,0) 1 = 1/)E r x)+e r x)) 1; subtracting 1 here accounts for the empty permutation. So, we begin with finding kernel shapes for all permutations in S 1.

6 133 TOUFIK MANSOUR The only shape obtained is ρ 1 = 1, and it is easy to see that sρ 1 ) = 1, cρ 1 ) = 0, fρ 1 ) =, X 1 ρ1 = Y 1 = { 1,0),0,1) }, X 1 ρ1 = Y1 = { 0,0),1,1) }, z ρ1 0,0) = z ρ1 1,0) = z ρ1 1,1) = z ρ1 0,1) = ) Therefore, 3.) fora = b = 0gives 1 E0 x)+e 0 x) ) 1 = xh 0 1,0)H 0 0,0)+xH 0 1,1)H 0 0,1) +xh 0 1,0)H 0 0,1)+xH 0 1,1)H 0 0,0); 3.6) 3.) fora = 1 and b = 0gives 1 E0 x) E 0 x) ) = xh0 0,0)+xH 0 0,1)+xH 0 1,0)+xH 0 1,1); 3.7) 3.) fora = 0 and b = 1gives 1 O0 x)+o 0 x) ) = xh 0 1,1)H 0 0,0)+xH 0 1,0)H 0 0,1) +xh 0 0,0)H 0 1,0)+xH 0 0,1)H 0 1,1); 3.8) and 3.) fora = b = 1gives 1 O0 x) O 0 x) ) = xh 0 0,1)H 0 0,0)+xH 0 1,1)H 0 1,0). 3.9) Our present aim is to find explicitly E 0 x) and O 0 x), thus we need the following notation. We define for all r 0. Clearly, M r x) = E r x) O r x), F r x) = E r x)+o r x) 3.10) H r 0,0) H r 0,1) = 1 Mr x)+m r x) ), H r 0,0)+H r 0,1) = 1 Fr x)+f r x) ), H r 1,0) H r 1,1) = 1 Mr x) M r x) ), 3.11) H r 1,0)+H r 1,1) = 1 Fr x) F r x) ), for all r 0. Therefore, by subtracting resp., adding) 3.8) and 3.6), and by subtracting resp., adding) 3.9) and3.7), we get M 0 x)+m 0 x) =, M 0 x) M 0 x) = x M 0 x)+m 0 x)), F 0 x)+f 0 x) = +x F 0 x) F 0 x)), F 0 x) F 0 x) = x F 0 x)+f 0 x)). 3.1)

7 COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION 1335 Hence, M 0 x) = x x, F 0 x) = 1 1 x. 3.13) x Theorem 3.3. i) The generating function for the number of even permutations avoiding 13 is given by see [16]) E 0 x) = x x ) 1 x. 3.1) x ii) The generating function for the number of odd permutations avoiding 13 is given by see [16]) O 0 x) = x x 1 1 ) 1 x. 3.15) x iii) The generating function for the number of permutations avoiding 13 is given by see [9]) F 0 x) = 1 1 x. 3.16) x 3.. The case r = 1. Since permutations in S do not exhibit kernel shapes distinct from ρ 1, the only possible new shape is the exceptional one, ρ = 13. Calculation of the parameters of ρ gives sρ ) = 3, cρ ) = 1, fρ ) = 3, X 1 ρ = Y 1 = { 1,0,0),0,1,0),0,0,1),1,1,1) }, X 1 ρ = Y1 = { 0,0,0),1,1,0),1,0,1),1,1,0) }, z ρ 0,0,0) = z ρ 1,0,0) = z ρ 0,1,0) = z ρ 1,1,0) = 1, z ρ 0,0,1) = z ρ 1,0,1) = z ρ 0,1,1) = z ρ 1,1,1) = ) Therefore, by Theorem 3.1, we have H 1 0,0) H 1 0,1) ) = M 1 x)+m 1 x) = x3 M0 x) M 0 x) ) M0 x)+m 0 x)), H 1 1,0) H 1 1,1) ) = M 1 x) M 1 x) = x M 0 x)m 1 x)+m 0 x)m 1 x) ) 3.18) x3 M0 x)+m 0 x) ) M0 x)+m 0 x)). Using the expression for M 0 x) see the case r = 0) we get M 1 x) = 1 1+3x +x ) + 1 3x x +x 3 1 x ) 1/. 3.19)

8 1336 TOUFIK MANSOUR Similarly, considering the expressions for H 1 0,0) + H 1 0,1) and H 1 1,0) + H 1 1,1) we get F 1 x) = 1 1 3x x 1)+ 1 x) 1/. 3.0) Theorem 3.. i) The generating function for the number of even permutations containing 13 exactly once is given by E 1 x) = 1 1 x x ) + 1 3x 1 x) 1/ + 1 3x x +x 3 1 x ) 3.1) 1/. ii) The generating function for the number of odd permutations containing 13 exactly once is given by O 1 x) = 1 x +x ) + 1 3x 1 x) 1/ 1 3x x +x 3 1 x ) 1/. 3.) iii) The generating function for the number of permutations containing 13 exactly once is given by see [6]) F 1 x) = 1 1 3x x 1)+ 1 x) 1/. 3.3) 3.3. The case r =. We have to check the kernel shapes of permutations in S. Exhaustive search adds four new shapes to the previous list; these are 13, 13, 13, and 13; besides, there is the exceptional 351 S 5. Calculation of the parameters s, c, f, z, X a, Y a is straightforward, and we get the following theorem. Theorem 3.5. i) The generating function for the number of even permutations containing 13 exactly twice is given by E x) = 1 x x 3 +3x x 1 ) + 1 x x 3 +9x 15x + ) 1 x) 3/ 1 16x 7 8x 6 76x 5 +6x +36x 3 1x 5x + ) 1 x ) 3.) 3/. ii) The generating function for the number of odd permutations containing 13 exactly once is given by O x) = 1 x +3x 3 5x x + ) + 1 x x 3 +9x 15x + ) 1 x) 3/ 3.5) x 7 8x 6 76x 5 +6x +36x 3 1x 5x + ) 1 x ) 3/. iii) The generating function for the number of permutations containing 13 exactly twice is given by see [1]) F x) = 1 x +3x ) + 1 x x 3 +9x 15x + ) 1 x) 3/. 3.6)

9 COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION The cases r = 3,,5,6. Let r = 3,,5,6; exhaustive search in S 6, S 8, S 10, and S 1 reveals 0, 10, 503, and 576 new nonexceptional kernel shapes, respectively, and we get the following theorem. Theorem 3.6. Let r = 3,,5,6, then M r x) = 1 Ar x)+b r x) 1 x ) r +1/), F r x) = 1 Cr x)+d r x)1 x) r +1/), 3.7) where A 3 x) = x 6 +10x 5 x 30x 3 +3x +7x, A x) = x 8 +1x 7 6x 6 90x x +85x 3 x 8x +1, A 5 x) = x x 9 76x 8 198x x 6 +0x 5 355x 171x 3 +6x +10x, A 6 x) = 56x 13 6x 1 618x x 10 10x x 8 +10x 7 170x 6 130x x +30x 3 88x 15x +, B 3 x) = 6x 11 30x x x 8 +11x 7 97x 6 5x 5 +31x +100x 3 3x 7x +, B x) = 56x x x x x 11 +x x x 8 838x 7 +77x x 5 775x 197x 3 +56x +8x 1, B 5 x) = 10x x x x x x x x x x x x x x 6 780x x +351x 3 98x 10x +, B 6 x) = 588x x x 399x x x x x x x x x x x x x x x x x 5 353x 63x x +15x, C 3 x) = x 3 5x +7x, C x) = 5x 7x 3 +x +8x 3,

10 1338 TOUFIK MANSOUR C 5 x) = 1x 5 17x +x 3 16x +1x, C 6 x) = x 6 x 5 +5x +x 3 0x +19x, D 3 x) = x 6 106x 5 +9x 30x x 7x +, D x) = x 9 +18x x 7 175x x x, D 5 x) = 50x x x x 8 166x x x x 893x 3 +50x 50x +, D 6 x) = x 1 +80x x x x x x x x x x 790x x 107x ) Moreover, for r = 3,,5,6, E r x) = 1 Ar x)+c r x)+d r x)1 x) r +1/ +B r x) 1 x ) r +1/), O r x) = 1 Ar x) C r x)+d r x)1 x) r +1/ B r x) 1 x ) 3.9) r +1/).. Further results and open questions. First of all, we simplify the expression L ρ r 1,...,r fρ) a,0) L ρ r 1,...,r fρ) a,1),.1) where a = 0,1, r j 0 for all j. Lemma.1. Let v {0,1} n be any vector and let a {1, 1}. Then n n n H r vj,x j H r vj,y j = a g r j),.) x Y a y Y a where g r j) = H r v j,0) H r v j,1) = 1/)M r x)+ 1) v j M r x)) for all j. Proof. For any two vectors u,v n, define uv = 1ifu i = v i for all i j and u j + v j = 1, otherwise uv = 0. Using the standard reflected Gray code, with each of the standard Gray-code vectors reflected left-for-right for more details, see [0]), we get that there exists an arrangement of the binary vectors of n,say0,...,0) = u 1,u,...,u n, such that the first m vectors in the sequence, say v 1,...,v m, when restricted to their first m coordinates, satisfy that v j v j+1 = 1 for all j. In such a context this arrangement is called Gray-code arrangement. Now we are ready to prove the lemma. Without loss of generality we can assume that 0,0,...,0) Ya n which means a = 1); otherwise it is enough to replace a by a. Let x 1,...,x n be all the vectors of n with the Gray-code arrangement. Thus, using 0,0,...,0) Ya n, we get that x i 1 Ya n and x i Y a n for all i = 1,,..., n 1. Therefore,

11 for all i = 1,,..., n 1, COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION 1339 n 1 i=1 n H r v j,x i 1 = n 1 i=1 = g r 1) j ) n H r v j,x i n 1) i 1 g r 1) H r v j,x i 1 n i=1 j= n 1 H r ṽ j,y i 1 j j j H r ṽ j,y i, n 1 j.3) where y p = x p,...,xp n ) for all p = 1,,..., n 1,andṽ = v,v 3,...,v n ). The Gray-code arrangement for x 1,...,x n implies that the vectors y 1,...,y n 1 are arranged as Graycode arrangement in n 1. Hence, by induction on n by definitions, the lemma holds for n = 1), we get that the expression equals a n g r j). As a remark, the vector 0,...,0) Y ρ z ρv) if and only if z ρv) = 1 for any kernel permutation ρ and vector v. Therefore, by Theorem 3.1 and Lemma.1 we get the following theorem. Theorem.. Let a {0,1} and r 0. Then 1 Mr x)+ 1) a M r x) ) δ r +a,0 = ρ K r +1 x sρ) r 1,...,r fρ) =r cρ) v X 1) a ρ) fρ) z ρ v) fρ) Mrj x)+ 1) v j M rj x) )..) As a remark, the above theorem yields two equations for a = 0 and a = 1) that are linear on M r x) and M r x). So,Theorem. provides a finite algorithm for finding M r x) for any given r 0, since we have to consider all permutations in S r +1 and to perform certain routine operations with all shapes found so far. Moreover, the amount of search can be decreased substantially due to the following proposition which holds immediately by Theorems 3. and.. Proposition.3. Let r 1, a {0,1}, and ρ = r 1,r +1,r 3,r,...,r j 3,r j,...,1,,..5) Then the expression x sρ) r 1,...,r fρ) =r cρ), r j 0 v X 1) a ρ) fρ) z ρ v) fρ) Mrj x)+ 1) v j M rj x) ).6)

12 130 TOUFIK MANSOUR is given by [r +)/] r + 1) j a+1 r x r +1 M 0 x) M 0 x) ) j M0 x)+m 0 x) ) r + j. j +1 a j=a.7) By this proposition, it is sufficient to search only permutations in S r. Besides, using Theorem. and the case r = 0, together with induction on r, we get the following result. Theorem.. M r x) is a rational function on x and 1 x for any r 0. In view of our explicit results, we have an even stronger conjecture. Conjecture.5. For any r 1, there exist polynomials A r x), B r x), C r x), and D r x) with integer coefficients such that E r x) = 1 Ar x)+b r x) ) + 1 C r x)1 x) r +1/ + 1 D r x) 1 x ) r +1/, O r x) = 1 Ar x) B r x) ) + 1 C r x)1 x) r +1/ 1 D r x) 1 x ).8) r +1/. Acknowledgments. The author is grateful to the anonymous referees for their helpful comments. This research was financed by EC s IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, Grant HPRN-CT References [1] N. Alon and E. Friedgut, On the number of permutations avoiding a given pattern, J. Combin. Theory Ser. A ), no. 1, [] M. D. Atkinson, Restricted permutations, Discrete Math ), no. 1 3, [3] M. Bóna, Exact enumeration of 13-avoiding permutations: a close link with labeled trees and planar maps, J. Combin. TheorySer. A ), no., [], The number of permutations with exactly r 13-subsequences is P-recursive in the size!, Adv. in Appl. Math ), no., [5], Permutations avoiding certain patterns: the case of length and some generalizations, Discrete Math ), no. 1 3, [6], Permutations with one or two 13-subsequences, Discrete Math ), no. 1 3, [7] T. Chow and J. West, Forbidden subsequences and Chebyshev polynomials, Discrete Math ), no. 1 3, [8] M. Fulmek, Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, Adv.inAppl.Math ), no., [9] D. E. Knuth,The Art of Computer Programming. Volume 3. Sorting and Searching, Addison- Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing, Massachusetts, [10] T. Mansour, Permutations containing and avoiding certain patterns, Formal Power Series and Algebraic Combinatorics Moscow, 000), Springer, Berlin, 000, pp [11] T. Mansour and A. Vainshtein, Restricted 13-avoiding permutations, Adv. in Appl. Math ), no. 3, [1], Counting occurrences of 13 in a permutation, Adv.inAppl.Math.8 00), no.,

13 COUNTING OCCURRENCES OF 13 IN AN EVEN PERMUTATION 131 [13] J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math ), no. 1 3, [1] J. Noonan and D. Zeilberger, The enumeration of permutations with a prescribed number of forbidden patterns, Adv. inappl. Math ), no., [15] A. Robertson, Permutations containing and avoiding 13 and 13 patterns, Discrete Math. Theor. Comput. Sci ), no., [16] R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin ), no., [17] Z. Stankova, Forbidden subsequences, Discrete Math ), no. 1 3, [18], Classification of forbidden subsequences of length, European J. Combin ), no. 5, [19] J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math ), no. 1 3, 7 6. [0] H. S. Wilf, Combinatorial Algorithms: An Update, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 55, Society for Industrial and Applied Mathematics SIAM), Pennsylvania, Toufik Mansour: Department of Mathematics, Chalmers University of Technology, 196 Göteborg, Sweden address: toufik@math.chalmers.se

14 Journal of Applied Mathematics and Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering Call for Papers As a multidisciplinary field, financial engineering is becoming increasingly important in today s economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems e.g., decision support systems, agent-based system, and web-based systems) This special issue will include but not be limited to) the following topics: Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation Authors should follow the Journal of Applied Mathematics and Decision Sciences manuscript format described at the journal site Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due December 1, 008 First Round of Reviews March 1, 009 Publication Date June 1, 009 Guest Editors Lean Yu, AcademyofMathematicsandSystemsScience, Chinese Academy of Sciences, Beijing , China; Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; yulean@amss.ac.cn Shouyang Wang, AcademyofMathematicsandSystems Science, Chinese Academy of Sciences, Beijing , China; sywang@amss.ac.cn K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk Hindawi Publishing Corporation