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Circular and Necklace Permutations

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Circular and Necklace Permutations Circular and Necklace Permutations Hiroshi KAJIMOTO and Mai OSABE Department of Mathematical Science, Faculty of Education Nagasaki University, Nagasaki 852-8521, Japan e-mail: [email protected] (Received October 31, 2006) Abstract We enumerate circular and necklace permutations which have beads of several colors, and count the related combinatorial numbers. 0 Introduction Circular and necklace permutations are standard subjects of high school com- binatorics. On the other hand they serve a good introduction to more ad- vanced subject of combinatorics, that is the P´olya theory of counting under group action. In fact these enumerations were historically a precursor of the theory [2, 4, p.558]. In this article we give the enumeration of circular and necklace permuations according to the given number of beads for each colors, and of the related combinatorial numbers. A circular permutation is defined to be an equivalence class of sequences under the cyclic group Cn, i.e., two sequences a1a2 : : : an and b1b2 : : : bn are equivalent if one is a cyclic shift of the other. A necklace (or necklace permu- tation) is defined to be an equivalence class of sequences under the dihedral group Dn, i.e., two sequences a1a2 : : : an and b1b2 : : : bn are equivalent if one is a cyclic shift of the other or if circular permutaions of them are mutually reflective with respect to some diameter. We make full use of the counting theory and look back the classical ex- amples, and show how to use P´olya's cycle index efficiently. We use combi- natorial notation, especially relative to partitions, as in [4]. 1 1 Cycle indices of cyclic and dihedral groups We begin with a brief accout of the counting theory under group action. A good introduction to the theory is [3, chapter 6], that is a lecture note without detailed proof. To those who want the proof see [1, chapter 5] or [4, section 7.24]. Let G be a subgroup of the full permutation group SX of a finite set X. Let n = jXj be the number of elements of X. Each element g 2 G is a permutation of X, so g is a product of several disjoint cycles of elements of X: g = (a1a2 : : : ak) ::: (c1c2 : : : cm). The cycle type of g determines a monomial b1(g) b2(g) bn(g) p1 p2 : : : pn , that is, g has b1(g) cycles of length 1, b2(g) cycles of length 2, . in the cycle decomposition, so (1b1(g)2b2(g) :::) ` n. Cycle index PG is a polynomial defined by 1 X P = pb1(g)pb2(g) : : : pbn(g): G j j 1 2 n G g2G Let R = fr; b; : : : ; wg be a set of colors. Then P´olya's main theorem is that all the inequivalent colorings f : X ! R under G are expressible by a i i i generating function PG substituted by pi = r + b + ··· + w . For circular and necklace permutations let X = f1; 2; : : : ng and let the n cyclic group Cn = haja = ei act on X in an obvious way: a = (1; 2; : : : ; n). The coloring f : X ! R is a sequence of length n and then Cn act on X the set R of sequences as a cyclic shift by a(r1r2 : : : rn) = r2r3 : : : rnr1. An equivalence class of the sequences under Cn is exactly a circular permutation. n 2 −1 The dihedral group Dn = ha; "ja = " = e; "a" = a i acts on X by putting " being a reflection with respect to some symmetry axis. For instance put ( − − b n c d n e (n)(1; n 1)(2; n 2) ::: ( 2 ; 2 ) when n odd " = − n n (1; n)(2; n 1) ::: ( 2 ; 2 + 1) when n even: An equivalence class of sequences under Dn is a necklace permutation. Lemma 1 Cycle indices of circular and necklace permutations are given by X 1 n=d PCn = '(d)pd ; n djn 8 (P ) < 1 n=d bn=2c djn '(d)pd + np1p2 when n odd P = 2n (P ) Dn : 1 n=d n n=2 2 (n=2)−1 2n djn '(d)pd + 2 (p2 + p1p2 ) when n even; where '(n) := #fd 2 Zjgcd(n; d) = 1; 1 ≤ d ≤ ng is the Euler function. 2 F For proof, notice that Cn = djn Cn(d) where Cn(d) is a subset of all the n=d ` elements of order d and that all elements in Cn(d) have the cycle typeF (d ) n and jCn(d)j = '(d). Notice also a coset decomposition: Dn = Cn "Cn and that all the elements of "Cn are reflections. When n is odd all the reflections bn=2c of "Cn have the cycle type (1)(2 ) ` n. When n is even we have two kinds n=2 of symmetry axes, half the reflections of "Cn have the cycle type (2 ) and another half the reflections have (12)(2n=2−1). The total number of circular and necklace permutations of length n with beads of k kinds of colors are given by substituting all pi = k in cycle indices. Corollary 2 The total number C(n; k) of circular permutations of length n with k colors of beads and N(n; k) of necklace permutations are ( ) 1 X 1 X n C(n; k) = '(d)kn=d = ' kd; n djn n djn d 0 ( 1 1 X nkdn=2e when n odd N(n; k) = @ '(d)kn=d + A : n (kn=2 + k(n=2)+1) when n even 2n djn 2 We here take the simplest: Example (n = 3). Cycle indices: 1 1 P = (p3 + 2p );P = (p3 + 2p + 3p p ): C3 3 1 3 D3 6 1 3 1 2 Total numbers: C(3; k) = (k3 + 2k)=3;N(3; k) = (k3 + 2k + 3k2)=6, k 1 2 3 4 5 C(3; k) 1 4 11 24 45 . N(3; k) 1 4 10 20 35 k = 3: We have beads of three colors, red r, blue b and white w. Substituting i i i the figure inventory pi = r + b + w in the cycle index, we get 1 n o P = (r + b + w)3 + 2(r3 + b3 + w3) C3 3 = r3 + b3 + w3 + r2b + r2w + b2r + b2w + w2r + w2b + 2rbw by the multinomial expansion. 3 r b w r3 + b3 + w3 r r b b w w b w r r r b r2b + r2w + b2r + b2w + w2r + w2b r r b w r b b b w w w w r r 2rbw w b b w The first 9 circles are self-reflective, the last 2 circles are pair-reflective each other, with respect to indicated symmetry axes. We can collect terms as X X X r3 = r3+b3+w3; r2b = r2b+r2w+b2r+b2w+w2r+w2b; rbw = rbw: P In these sums, for instance r2b indicates that terms with two beads of one 2 color and one bead of the second color are summedP up, r b is the typical term of the possible( ) color choices. The sum rbw has just one term when k ≥ k = 3, but has 3 terms when k 3 and has no terms hence = 0 when k < 3. Then we have Xk Xk Xk 3 2 PC3 = r + r b + 2 rbw: This expression is independent of the indicated number k of colors. In this example we easily have for necklaces, Xk Xk Xk 3 2 PD3 = r + r b + rbw 4 P λ by computation or by inspection. We will seek their coefficient [ r ]PG, which is the number of essentially inequivalent color patterns under group action. 2 Circular permutations To get all the circular permutations of length n with beads of k kinds of d d ··· d color, we have to expand PCn by substituting pd = r1 + r2 + + rk, the power sum symmetric polyomials. In this case the expansion is fairly easy. Remember the multinomial expansion and write that in the following way: ! ! X n X n Xk (x + x + ··· + x )n = xα = xµ: 1 2 k α µ jαj=n,α2Nk µ`n;l(µ)≤k P Here l(µ) is the length of a partition µ and k xµ is a monomial symmetric polynomial mµ of k variables [4, section7.3] defined by Xk X µ α x = mµ = x α where the last sum runs over all distinct permutations α = (α1; α2;:::) of the Pentries of the partition µ = (µ1; µ2;:::). A monomial symmetric polynomial k xµ collects all the colorings whose choices of the numbers of different k colors of beads are according to the partition µ ` n. We have by the substitution 0 ! 1 X X X Xk 1 n=d 1 @ n=d dµA PCn = '(d)pd = '(d) r n djn n djn µ`n=d µ where dµ = (dµ1; dµ2;:::) ` djµj. Putting λ = dµ ` n, we get the generating function of circular permutations of length n with k colors of beads: Theorem 3 8 ! 9 X < 1 X n=d = Xk P = '(d) rλ: Cn :n λ/d ; λ`n djgcdfλ1,λ2;:::g where λ/d = (λ1=d; λ2=d; : : :) ` n=d. 5 P P ( ) k λ n=d The coefficient [ r ]PCn = djgcdfλ1,λ2;:::g λ/d '(d)=n is the number of es- sentially different circular permutations according to the number of each color indicated by the partition λ ` n. For instance the number of circular permu- tations ofP length n = 6 with two( ) red beads,( ) two blue beads and two white 2 2 2 f 6 3 g beads is [ r b w ]PC6 = '(1) 23 +'(2) 13 =6 = 16.
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