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Palindromic permutations and generalized Smarandache palindromic permutations.

Abstract The idea of left(right) palindromic permutations(LPPs, RPPs) and left(right) generalized Smarandache palindromic permutations(LGSPPs, RGSPPs) are introduced in symmetric groups [S.sub.n] of degree n. It is shown that in [S.sub.n], there exist a LPP and a RPP and they are unique(this fact is demonstrated using [S.sub.2] and [S.sub.3]). The dihedral group [D.sub.n] is shown to be generated by a RGSPP and a LGSPP(this is observed to be true in [S.sub.3]) but the geometric interpretations of a RGSPP and a LGSPP are found not to be rotation and reflection respectively. In [S.sub.3], each permutation is at least a RGSPP or a LGSPP. There are 4 RGSPPs and 4 LGSPPs in [S.sub.3], while 2 permutations are both RGSPPs and LGSPPs. A permutation in [S.sub.n] is shown to be a LPP or RPP(LGSPP or RGSPP) if and only if its inverse is a LPP or RPP(LGSPP or RGSPP) respectively. Problems for future studies are raised.

Keywords Permutation, symmetric groups, palindromic permutations, generalized Smarandache palindromic permutations.

[section]1. Introduction

According to Ashbacher and Neirynck [1], an integer is said to be a palindrome if it reads the same forwards and backwards. For example, 12321 is a palindromic number. They also stated that it is easy to prove that the density of the palindromes is zero in the set of positive integers and they went ahead to answer the question on the density of generalized Smarandache palindromes (GSPs) by showing that the density of GSPs in the positive integers is approximately 0:11. Gregory [2], Smarandache [8] and Ramsharan [7] defined generalized Smarandache palindrome (GSP) as any integer or number of the form

[a.sub.1][a.sub.2][a.sub.3] ... [a.sub.n][a.sub.n] ... [a.sub.3][a.sub.2][a.sub.1] or [a.sub.1][a.sub.2][a.sub.3] ... [a.sub.n-1][a.sub.n][a.sub.n-1] ... [a.sub.3][a.sub.2][a.sub.1]

where all [a.sub.1], [a.sub.2], [a.sub.3], ..., [a.sub.n] [member of] N having one or more digits. On the other hand, Hu [3] calls any integer or number of this form a Smarandache generalized palindrome (SGP). His naming will not be used here the first naming will be adopted. Numbers of this form have also been considered by Khoshnevisan [4], [5] and [6]. For the sake of clarification, it must be mentioned that the possibility of the trivial case of enclosing the entire number is excluded. For example, 12345 can be written as (12345). In this case, the number is simply said to be a palindrome or a palindromic number as it was mentioned earlier on. So, every number is a GSP. But this possibility is eliminated by requiring that each number be split into at least two segments if it is not a regular palindrome. Trivially, since each regular palindrome is also a GSP and there are GSPs that are not regular palindromes, there are more GSPs than there are regular palindromes. As mentioned by Gregory [2], very interesting GSPs are formed from smarandacheian sequences. For an illustration he cited the smarandacheian sequence

11, 1221, 123321, ..., 123456789987654321, 1234567891010987654321, 12345678910111110987654321, ...

and observed that all terms are all GSPs. He also mentioned that it has been proved that the GSP 1234567891010987654321 is a prime and concluded his work by possing the question of How many primes are in the GSP sequence above?

Special mappings such as morphisms(homomorphisms, endomorphisms, automorphisms, isomorphisms e.t.c) have been useful in the study of the properties of most algebraic structures(e.g groupoids, quasigroups, loops, semigroups, groups e.tc.). In this work, the notion of palindromic permutations and generalized Smarandache palindromic permutations are introduced and studied using the symmetric group on the set N and this can now be viewed as the study of some palindromes and generalized Smarandache palindromes of numbers.

The idea of left(right) palindromic permutations(LPPs, RPPs) and left(right) generalized Smarandache palindromic permutations(LGSPPs, RGSPPs) are introduced in symmetric groups [S.sub.n] of degree n. It is shown that in [S.sub.n], there exist a LPP and a RPP and they are unique. The dihedral group [D.sub.n] is shown to be generated by a RGSPP and a LGSPP but the geometric interpretations of a RGSPP and a LGSPP are found not to be rotation and reflection respectively. In [S.sub.3], each permutation is at least a RGSPP or a LGSPP. There are 4 RGSPPs and 4 LGSPPs in [S.sub.3], while 2 permutations are both RGSPPs and LGSPPs. A permutation in [S.sub.n] is shown to be a LPP or RPP(LGSPP or RGSPP) if and only if its inverse is a LPP or RPP(LGSPP or RGSPP) respectively. Some of these results are demonstrated with [S.sub.2] and [S.sub.3]. Problems for future studies are raised.

But before then, some definitions and basic results on symmetric groups in classical group theory which shall be employed and used are highlighted first.

[section]2. Preliminaries

Definition 2.1 Let X be a non-empty set. The group of all permutations of X under composition of mappings is called the symmetric group on X and is denoted by [S.sub.X]. A subgroup of [S.sub.X] is called a permutation group on X.

It is easily seen that a bijection X [equivalent] Y induces in a natural way an isomorphism [S.sub.X] [congruent to] [S.sub.Y]. If |X| = n, [S.sub.X] is denoted by [S.sub.n] and called the symmetric group of degree n.

A permutation [sigma] [member of] [S.sub.n] can be exhibited in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

consisting of two rows of integers; the top row has integers 1, 2, ..., n, usually (but not necessarily) in their natural order, and the bottom row has [sigma](i) below i for each i = 1, 2, ..., n. This is called a two-row notation for a permutation. There is a simpler, one-row notation for a special kind of permutation called cycle.

Definition 2.2 Let [sigma] [member of] [S.sub.n]. If there exists a list of distinct integers [x.sub.1], ..., [x.sub.r] [member of] N such that

[sigma]([x.sub.i]) = [x.sub.i+1], i = 1, ..., r - 1, [sigma]([x.sub.r]) = [x.sub.1], [sigma](x) = x if x [??] {[x.sub.1], ..., [x.sub.r]},

then [sigma] is called a cycle of length r and denoted by ([x.sub.1] ... [x.sub.r]).

Remark 2.1 A cycle of length 2 is called a transposition. In other words, a cycle ([x.sub.1] ... [x.sub.r]) moves the integers ([x.sub.1] ... [x.sub.r]) one step around a circle and leaves every other integer in N. If [sigma](x) = x, we say [sigma] does not move x. Trivially, any cycle of length 1 is the identity mapping I or e. Note that the one-row notation for a cycle does not indicate the degree n, which has to be understood from the context.

Definition 2.3 Let X be a set of points in space, so that the distance d(x, y) between points x and y is given for all x, y [member of] X. A permutation [sigma] of X is called a symmetry of X if

d([sigma](x), [sigma](y)) = d(x, y) [for all] x, y [member of] X.

Let X be the set of points on the vertices of a regular polygon which are labelled 1, 2, ..., n i.e X = {1, 2, ..., n}.

** The group of symmetries of a regular polygon [P.sub.n] of n sides is called the dihedral group of degree n and denoted [D.sub.n].

Remark 2.2 It must be noted that [D.sub.n] is a subgroup of [S.sub.n] i.e [D.sub.n] [less than or equal to] [S.sub.n]. Definition 2.4 Let [S.sub.n] be a symmetric group of degree n. If [sigma] [member of] [S.sub.n] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

1. the number [N.sub.[lambda]]([sigma]) = 12 ... n[sigma](n) ... [sigma](1) is called the left palindromic value(LPV) of [sigma].

2. the number [N.sub.[rho]]([sigma]) = 12 ... n[sigma](1) ... [sigma](n) is called the right palindromic value(RPV) of [sigma].

Definition 2.5 Let [sigma] [member of] [S.sub.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If X = N, then

1. [sigma] is called a left palindromic permutation(LPP) if and only if the number [N.sub.[lambda]]([sigma]) is a palindrome.

P[P.sub.[lambda]]([S.sub.X]) = {[sigma] [member of] [S.sub.X]: [sigma] is a LPP}

2. [sigma] is called a right palindromic permutation(RPP) if and only if the number [N.sub.[rho]]([sigma]) is a palindrome.

P[P.sub.[rho]]([S.sub.X]) = { [sigma] [member of] [S.sub.X]: [sigma] is a RPP}

3. [sigma] is called a palindromic permutation(PP) if and only if it is both a LPP and a RPP.

PP([S.sub.X]) = {[sigma] [member of] [S.sub.X]: [sigma] is a LPP and a RPP} = P[P.sub.[lambda]]([S.sub.X]) [intersection] P[P.sub.[rho]]([S.sub.X])

Definition 2.6 Let [sigma] [member of] [S.sub.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If X = N, then

1. [sigma] is called a left generalized Smarandache palindromic permutation(LGSPP) if and only if the number [N.sub.[lambda]] is a GSP.

GSP[P.sub.[lambda]]([S.sub.X]) = {[sigma] [member of] [S.sub.X]: [sigma] is a LGSPP}

2. [sigma] is called a right generalized Smarandache palindromic permutation(RGSPP) if and only if the number [N.sub.[rho]] is a GSP.

GSP[P.sub.[rho]]([S.sub.X]) = {[sigma] [member of] [S.sub.X]: [sigma] is a RGSPP}

3. [sigma] is called a generalized Smarandache palindromic permutation(GSPP) if and only if it is both a LGSPP and a RGSPP.

GSPP([S.sub.X]) = {[sigma] [member of] [S.sub.X]: [sigma] is a LGSPP and a RGSPP} = GSP[P.sub.[lambda]]([S.sub.X]) [intersection] GSP[P.sub.[rho]]([S.sub.X])

Theorem 2.1 (Cayley Theorem) Every group is isomorphic to a permutation group.

Theorem 2.2 The dihedral group [D.sub.n] is a group of order 2n generated by two elements [sigma], [tau] satisfying [[sigma].sup.n] = e = [[tau].sup.2] and [tau][sigma] = [[sigma].sup.n-1][tau], where

[sigma] = (1 2 ... n),

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]3. Main Results

Theorem 3.1 In any symmetric group [S.sub.n] of degree n, there exists

1. a LPP and it is unique.

2. a RPP and it is unique.

But there does not exist a PP.

Proof Let [sigma] [member of] [S.sub.n], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1. When

[sigma](n) = n, [sigma](n - 1) = n - 1, ..., [sigma](2) = 2, [sigma](1) = 1

then the number

[N.sub.[lambda]]([sigma]) = 12 ... n[sigma](n) ... [sigma](2)[sigma](1) = 12 ... nn ... 21

is a palindrome which implies [sigma] [member of] P[P.sub.[lambda]]([S.sub.n]). So there exist a LPP. The uniqueness is as follows. Observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [S.sub.n] is a group for all n [member of] N and I is the identity element (mapping), then it must be unique.

2. When

[sigma](1) = n, [sigma](2) = n - 1, ..., [sigma](n - 1) = 2, [sigma](n) = 1,

then the number

[N.sub.[rho]]([sigma]) = 12 ... n[sigma](1) ... [sigma](n - 1)[sigma](n) = 12 ... nn ... 21

is a palindrome which implies [sigma] [member of] P[P.sub.[rho]][S.sub.n]. So there exist a RPP. The uniqueness is as follows. If there exist two of such, say [[sigma].sub.1] and [[sigma].sub.2] in [S.sub.n], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that

[N.sub.[rho]]([[sigma].sub.1]) = 12 ... n[[sigma].sub.1](1) ... [[sigma].sub.1](n - 1)[[sigma].sub.1](n)

and

[N.sub.[rho]]([[sigma].sub.2]) = 12 ... n[[sigma].sub.2](1) ... [[sigma].sub.2](n - 1)[[sigma].sub.2](n)

are palindromes which implies

[[sigma].sub.1](1) = n, [[sigma].sub.1](2) = n - 1, ..., [[sigma].sub.1](n - 1) = 2, [[sigma].sub.1](n) = 1,

and

[[sigma].sub.2](1) = n, [[sigma].sub.2](2) = n - 1, ..., [[sigma].sub.2](n - 1) = 2, [[sigma].sub.2](n) = 1.

So, [[sigma].sub.1] = [[sigma].sub.2], thus [sigma] is unique.

The proof of the last part is as follows. Let us assume by contradiction that there exists a PP [sigma] [member of] [S.sub.n]. Then if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[N.sub.[rho]]([sigma]) = 12 ... n[sigma](1) ... [sigma](n - 1)[sigma](n)

are palindromes. So that [sigma] [member of] [S.sub.n] is a PP. Consequently,

n = [sigma](n) = 1, n - 1 = [sigma](n - 1) = 2, ..., 1 = [sigma](1) = n,

so that [sigma] is not a bijection which means [sigma] [??] [S.sub.n]. This is a contradiction. Hence, no PP exist.

Example 3.1 Let us consider the symmetric group [S.sub.2] of degree 2. There are two permutations of the set {1, 2} given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So, I and [delta] are both RGSPPs and LGSPPs which implies I and [delta] are GSPPs i.e I, [delta] [member of] GSP[P.sub.[rho]]([S.sub.2]) and I, [delta] [member of] GSP[P.sub.[rho]]([S.sub.2]) [??] I, [delta] [member of] GSPP([S.sub.2]). Therefore, GSPP([S.sub.2]) = [S.sub.2]. Furthermore, it can be seen that the result in Theorem 3.1 is true for [S.sub.2] because only I is a LPP and only [delta] is a RPP. There is no PP as the theorem says.

Example 3.2 Let us consider the symmetric group [S.sub.3] of degree 3. There are six permutations of the set {1,2,3} given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As claimed in Theorem 3.1, the unique LPP in [S.sub.3] is I while the unique RPP in [S.sub.3] is [[tau].sub.2]. There is no PP as the theorem says.

Lemma 3.1 In [S.sub.3], the following are true.

1. At least [sigma] [member of] GSP[P.sub.[rho]]([S.sub.3]) or [sigma] [member of] GSP[P.sub.[lambda]]([S.sub.3]) [for all] [member of] 2 [S.sub.3].

2. |GSP[P.sub.[rho]]([S.sub.3])| = 4, |GSP[P.sub.[lambda]]([S.sub.3])| = 4 and |GSPP([S.sub.3])| = 2.

Proof Observe the following:

[N.sub.[lambda]](I) = 123321, [N.sub.[rho]](I) = 123123 = (123)(123).

[N.sub.[lambda]]([[sigma].sub.1]) = 123132, [N.sub.[rho]]([[sigma].sub.1]) = 123231 = 1(23)(23)1.

[N.sub.[lambda]]([[sigma].sub.2]) = 123213, [N.sub.[rho]]([[sigma].sub.2]) = 123312 = (12)(33)(12).

[N.sub.[lambda]]([[tau].sub.1]) = 123231 = 1(23)(23)1, [N.sub.[rho]]([[tau].sub.1]) = 123132.

[N.sub.[lambda]]([[tau].sub.2]) = 123123 = (123)(123), [N.sub.[rho]]([[tau].sub.2]) = 123321.

[N.sub.[lambda]]([[tau].sub.3]) = 123312 = (12)(33)(12), [N.sub.[rho]]([[tau].sub.3]) = 123213.

So, GSP[P.sub.[lambda]]([S.sub.3]) = {I, [[tau].sub.1], [[tau].sub.2], [[tau].sub.3]} and GSP[P.sub.[rho]]([S.sub.3]) = {I, [[sigma].sub.1], [[sigma].sub.2], [[tau].sub.2]}. Thus, 1. is true .Therefore, |GSP[P.sub.[rho]]([S.sub.3])| = 4, |GSP[P.sub.[lambda]]([S.sub.3])| = 4 and |GSPP([S.sub.3])| = |GSP[P.sub.[rho]]([S.sub.3])[intersection]GSP[P.sub.[lambda]]([S.sub.3])| = 2. So, 2. is true.

Lemma 3.2 [S.sub.3] is generated by a RGSPP and a LGSPP.

Proof Recall from Example 3.2 that

[S.sub.3] = {I, e, [[sigma].sub.1], [[sigma].sub.2], [[tau].sub.1], [[tau].sub.2], [[tau].sub.3]}.

If [sigma] = [[sigma].sub.1] and [tau] = [[tau].sub.1], then it is easy to verify that

[[sigma].sup.2] = [[sigma].sub.2], [[sigma].sup.3] = e, [[tau].sup.2] = e, [sigma][tau] = [[tau].sub.3], [[sigma].sup.2][tau] = [[tau].sub.2] = [tau][sigma] hence,

[S.sub.3] = {e, [sigma], [[sigma].sup.2], [tau], [sigma][tau], [[sigma].sup.2][[tau].sub.3]} [??] [S.sub.3] = <[sigma], [tau]>.

From the proof Lemma 3.1, [sigma] is a RGSPP and [tau] is a LGSPP. This justifies the claim.

Remark 3.1 In Lemma 3.2, [S.sub.3] is generated by a RGSPP and a LGSPP. Could this statement be true for all [S.sub.n] of degree n? Or could it be true for some subgroups of [S.sub.n]? Also, it is interesting to know the geometric meaning of a RGSPP and a LGSPP. So two questions are possed and the two are answered.

Question 3.1 1. Is the symmetric group [S.sub.n] of degree n generated by a RGSPP and a LGSPP? If not, what permutation group (s) is generated by a RGSPP and a LGSPP?

2. Are the geometric interpretations of a RGSPP and a LGSPP rotation and reflection respectively?

Theorem 3.2 The dihedral group [D.sub.n] is generated by a RGSPP and a LGSPP i.e [D.sub.n] = <[sigma], [tau]> where [sigma] [member of] GSP[P.sub.[rho]]([S.sub.n]) and [tau] [member of] GSP[P.sub.[lambda]]([S.sub.n]).

Proof Recall from Theorem 2.2 that the dihedral group [D.sub.n] = <[sigma], [tau]> where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe that

[N.sub.[rho]]([sigma]) = 123 ... n23 ... n1 = 1(23 ...n)(23 ... n)1, [N.sub.[lambda]]([sigma]) = 123 ... n1n ... 32.

[N.sub.[rho]]([tau]) = 12 ... n1n ... 2, [N.sub.[lambda]]([tau]) = 12 ... n2 ... n1 = 1(2 ... n)(2 ... n)1.

So, [sigma] [member of] GSP[P.sub.[rho]]([S.sub.n]) and [tau] [member of] GSP[P.sub.[lambda]]([S.sub.n]). Therefore, the dihedral group [D.sub.n] is generated by a RGSPP and a LGSPP.

Remark 3.2 In Lemma 3.2, it was shown that [S.sub.3] is generated by a RGSPP and a LGSPP. Considering Theorem 3.2 when n = 3, it can be deduced that [D.sub.3] will be generated by a RGSPP and a LGSPP. Recall that |[D.sub.3]| = 2 x 3 = 6, so [S.sub.3] = [D.sub.3]. Thus Theorem 3.2 generalizes Lemma 3.2.

Rotations and Reflections Geometrically, in Theorem 3.2, [sigma] is a rotation of the regular polygon [P.sub.n] through an angle 2[pi]/n in its own plane, and [tau] is a reflection (or a turning over) in the diameter through the vertex 1. It looks like a RGSPP and a LGSPP are formed by rotation and reflection respectively. But there is a contradiction in [S.sub.4] which can be traced from a subgroup of [S.sub.4] particularly the Klein four-group. The Klein four-group is the group of symmetries of a four sided non-regular polygon(rectangle). The elements are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe the following:

[N.sub.[rho]]([[delta].sub.1]) = 12343412 = (12)(34)(34)(12), [N.sub.[lambda]]([[delta].sub.1]) = 12342143.

[N.sub.[rho]]([[delta].sub.2]) = 12342143, [N.sub.[lambda]]([[delta].sub.2]) = 12343412 = (12)(34)(34)(12).

[N.sub.[rho]]([[delta].sub.3]) = 12344321 = 123(44)321, [N.sub.[lambda]]([[delta].sub.3]) = 12341234 = (1234)(1234).

So, [[delta].sub.1] is a RGSPP while [[delta].sub.2] is a LGSPP and [[delta].sub.3] is a GSPP. Geometrically, [[delta].sub.1] is a rotation through an angle of [pi] while [[delta].sub.2] and [[delta].sub.3] are reflections in the axes of symmetry parallel to the sides. Thus [[delta].sub.3] which is a GSPP is both a reflection and a rotation, which is impossible. Therefore, the geometric meaning of a RGSPP and a LGSPP are not rotation and reflection respectively. It is difficult to really ascertain the geometric meaning of a RGSPP and a LGSPP if at all it exist.

How beautiful will it be if GSP[P.sub.[rho]]([S.sub.n]), P[P.sub.[rho]]([S.sub.n]), GSP[P.sub.[lambda]]([S.sub.n]), P[P.sub.[lambda]]([S.sub.n]), GSPP([S.sub.n]) and PP([S.sub.n]) form algebraic structures under the operation of map composition.

Theorem 3.3 Let [S.sub.n] be a symmetric group of degree n. If [sigma] [member of] [S.sub.n], then

1. [sigma] [member of] P[P.sub.[lambda]]([S.sub.n]) [??][[sigma].sup.-1] [member of] P[P.sub.[lambda]]([S.sub.n]).

2. [sigma] [member of] P[P.sub.[rho]]([S.sub.n]) [??][[sigma].sup.-1] [member of] P[P.sub.[rho]]([S.sub.n]).

3. I [member of] P[P.sub.[lambda]]([S.sub.n]).

Proof 1. [sigma] [member of] P[P.sub.[lambda]]([S.sub.n]) implies

[N.sub.[lambda]]([sigma]) = 12 ... n[sigma](n) ... [sigma](2)[sigma](1)

is a palindrome. Consequently,

[sigma](n) = n, [sigma](n - 1) = n - 1, ..., [sigma](2) = 2, [sigma](1) = 1.

So

[N.sub.[lambda]]([[sigma].sup.-1]) = [sigma](1)[sigma](2) ... [sigma](n)n ... 21 = 12 ... nn ... 21 [??] [[sigma].sup.- 1] [member of] P[P.sub.[lambda]]([S.sub.n]).

The converse is similarly proved by carrying out the reverse of the procedure above. 2. [sigma] [member of] P[P.sub.[rho]]([S.sub.n]) implies

[N.sub.[rho]]([sigma]) = 12 ... n[sigma](1) ... [sigma](n - 1)[sigma](n)

is palindrome. Consequently,

[sigma](1) = n, [sigma](2) = n - 1, ..., [sigma](n - 1) = 2, [sigma](n) = 1.

So

[N.sub.[rho]](1) = [sigma](1)[sigma](2) ... [sigma](n - 1)[sigma](n)12 ... n = n ... 2112 ...n [??] [[sigma].sup.-1] [member of] P[P.sub.[rho]]([S.sub.n]).

The converse is similarly proved by carrying out the reverse of the procedure above. 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[N.sub.[lambda]](I) = 12 ... nn ... 21 [??] I [member of] P[P.sub.[lambda]]([S.sub.n]).

Theorem 3.4 Let [S.sub.n] be a symmetric group of degree n. If [sigma] [member of] [S.sub.n], then

1.[sigma] [member of] GSP[P.sub.[lambda]]([S.sub.n]) [??][[sigma].sup.-1] [member of] GSP[P.sub.[lambda]]([S.sub.n]).

2.[sigma] [member of] GSP[P.sub.[rho]]([S.sub.n]) [??][[sigma].sup.-1] [member of] GSP[P.sub.[rho]]([S.sub.n]).

3.I [member of] GSPP([S.sub.n]).

Proof If [sigma] [member of] [S.sub.n], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So

[N.sub.[lambda]]([sigma]) = 12 ... n[sigma](n) ... [sigma](2)[sigma](1)

and

[N.sub.[rho]]([sigma]) = 12 ... n[sigma](1) ... [sigma](n - 1)[sigma](n)

are numbers with even number of digits whether n is an even or odd number. Thus, [N.sub.[rho]]([sigma]) and [N.sub.[lambda]]([sigma]) are GSPs defined by

[a.sub.1][a.sub.2][a.sub.3] ... [a.sub.n][a.sub.n] ... [a.sub.3][a.sub.2][a.sub.1]

and not

[a.sub.1][a.sub.2][a.sub.3] ... [a.sub.n-1][a.sub.n][a.sub.n-1] ... [a.sub.3][a.sub.2][a.sub.1]

where all [a.sub.1], [a.sub.2], [a.sub.3], ..., [a.sub.n] [member of] N having one or more digits because the first has even number of digits (or grouped digits) while the second has odd number of digits (or grouped digits). The following grouping notations will be used:

[([a.sub.i]).sup.n.sub.i=1] = [a.sub.1][a.sub.2][a.sub.3] ... [a.sub.n]

and

[[[a.sub.i]].sup.n.sub.i=1] = [a.sub.n][a.sub.n-1][a.sub.n-2] ... [a.sub.3][a.sub.2][a.sub.1].

Let [sigma] [member of] [S.sub.n] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [x.sub.i] [member of] N, [for all] i [member of] N.

1. So [sigma] [member of] GSP[P.sub.[lambda]]([S.sub.n]) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a GSP, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [n.sub.n] = n. The interval of integers [1, n] is partitioned into

[1, n] = [1, [n.sub.1]] [union] [[n.sub.1] + 1, [n.sub.2]] [union] ... [union] [[n.sub.n-2] + 1, [n.sub.n-1]] [union] [[n.sub.n-1], [n.sub.n]].

The length of each grouping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined by the corresponding interval of integers [[n.sub.i] + 1, [n.sub.i+1]] and it is a matter of choice in other to make the number [N.sub.[lambda]]([sigma]) a GSP.

Now that [N.sub.[lambda]]([sigma]) is a GSP, the following are true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a GSP. Hence, [[sigma].sup.-1] [member of] GSP[P.sub.[lambda]]([S.sub.n]).

The converse can be proved in a similar way since [([[sigma].sup.-1]).sup.-1] = [sigma].

2. Also, [sigma] [member of] GSP[P.sub.[rho]]([S.sub.n]) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a GSP, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [n.sub.n] = n. The interval [1, n] is partitioned into

[1, n] = [1, [n.sub.1]] [union] [[n.sub.1] + 1, [n.sub.2]] [union] ... [union] [[n.sub.n-2] + 1, [n.sub.n-1]] [union] [[n.sub.n-1], [n.sub.n]].

The length of each grouping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is determined by the corresponding interval of integers [[n.sub.i] + 1, [n.sub.i+1]] and it is a matter of choice in other to make the number [N.sub.[rho]]([sigma]) a GSP.

Now that [N.sub.[rho]]([sigma]) is a GSP, the following are true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a GSP. Hence, [[sigma].sup.-1] [member of] GSP[P.sub.[rho]]([S.sub.n]).

The converse can be proved in a similar way since [([[sigma].sup.-1]).sup.-1] = [sigma]. 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[N.sub.[lambda]](I) = 12 ... nn ... 21 = 12 ... (nn) ... 21 [??] I [member of] GSP[P.sub.[lambda]]([S.sub.n]) and

[N.sub.[rho]](I) = (12 ... n)(12 ... n) [??] I [member of] GSP[P.sub.[rho]]([S.sub.n])

thus I [member of] GSPP([S.sub.n]).

[section]4. Conclusion and Future studies

By Theorem 3.1, it is certainly true in every symmetric group [S.sub.n] of degree n there exist at least a RGSPP and a LGSPP(although they are actually RPP and LPP). Following Example 3.1, there are 2 RGSPPs, 2 LGSPPs and 2 GSPPs in [S.sub.2] while from Lemma 3.1, there are 4 RGSPPs, 4 LGSPPs and 2 GSPPs in [S.sub.3]. Also, it can be observed that

|GSP[P.sub.[rho]]([S.sub.2])| + |GSP[P.sub.[lambda]]([S.sub.2])| - |GSPP([S.sub.2])| = 2! = |[S.sub.2]|

and

|GSP[P.sub.[rho]]([S.sub.3])| + |GSP[P.sub.[lambda]]([S.sub.3])| - |GSPP([S.sub.3])| = 3! = |[S.sub.3]|.

The following problems are open for further studies.

Problem 4.1 1. How many RGSPPs, LGSPPs and GSPPs are in [S.sub.n]?

2. Does there exist functions [f.sub.1], [f.sub.2], [f.sub.3]: N [right arrow] N, such that |GSP[P.sub.[rho]]([S.sub.n])| = [f.sub.1](n), |GSP[P.sub.[lambda]]([S.sub.n])| = [f.sub.2](n) and |GSPP([S.sub.n])| = [f.sub.3](n)?

3. In general, does the formula

|GSP[P.sub.[rho]]([S.sub.n])| + |GSP[P.sub.[lambda]]([S.sub.n])| - |GSPP([S.sub.n])| = n! = |[S.sub.n]|?

hold. If not, for what other n > 3 is it true?

The GAP package or any other appropriate mathematical package could be helpful in investigating the solutions to them.

If the first question is answered, then the number of palindromes that can be formed from the set {1, 2, ..., n} can be known since in the elements of [S.sub.n], the bottom row gives all possible permutation of the integers 1, 2, ..., n.

The Cayley Theorem(Theorem 2.1) can also be used to make a further study on generalized Smarandache palindromic permutations. In this work, N was the focus and it does not contain the integer zero. This weakness can be strengthened by considering the set [Z.sub.n] = {0, 1, 2, ..., n - 1}, [for all] n [member of] N. Recall that ([Z.sub.n], +) is a group and so by Theorem 2.1 ([Z.sub.n], +) is isomorphic to a permutation group particularly, one can consider a subgroup of the symmetric group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

[1] C. Ashbacher, L. Neirynck, The density of generalized Smarandache palindromes, http://www.gallup.unm.edu/~smarandache/GeneralizedPalindromes.htm.

[2] G. Gregory, Generalized Smarandache palindromes, http://www.gallup.unm.edu/~smarandache/GSP.htm.

[3] C. Hu (2000), On Smarandache generalized palindrome, Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, University of Craiova, Craiova Romania, atlas-conferences.com/c/a/f/t/25.htm.

[4] M.Khoshnevisan, Manuscript, 2003.

[5] M. Khoshnevisan, Generalized Smarandache palindrome, Mathematics Magazine, Aurora, Canada, 2003.

[6] M. Khoshnevisan, Proposed problem 1062, The PME Journal, USA, 11(2003), pp. 501.

[7] K. Ramsharan, Manusript, 2003. www.research.att.com/~njas/sequences/A082461.

[8] F. Smarandache, Sequences of Numbers Involved in unsolved problems, http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm, 2006, pp. 141.

Temitope Gbolahan Jaiyeola (1)

Department of Mathematics, Obafemi Awolowo University, Ile Ife, Nigeria.
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Date:Sep 1, 2006
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