A Note on Modular Partitions and Necklaces

Transcription

1 A Note on Moular Partitions an Neclaces N. J. A. Sloane, Rutgers University an The OEIS Founation Inc. South Aelaie Avenue, Highlan Par, NJ 08904, USA. May 6, 204 Abstract Following Jens Voß, let T (n, be the number of moular partitions of n into parts, that is, the number of -tuples (u, u 2,..., u with 0 u u 2 u n such that j u j 0 mo n. The purpose of this note is to use Molien s theorem to show that T (n, is also equal to the number of bi-colore neclaces with n beas of one color an beas of another color. Introuction Following Jens Voß [9], let T (n, be the number of -tuples u = (u, u 2,..., u with 0 u u 2 u n such that j u j 0 mo n. State another way, T (n, is the number of ways to write 0 as a sum of elements of Z/nZ. Voß calls u a moular partition of n into parts. He compute the numbers T (n, for n + 20, an part of his table is shown here (the rows correspon to n = 0,, 2,..., 0 an the columns to = 0,, 2,...:,,,,,,,,,,,,,,,,,,,,...,,,,,,,,,,,,,,,,,,,,...,,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,0,...,,2,4,5,7,0,2,5,9,22,26,3,35,40,46,5,57,...,,3,5,0,4,22,30,43,55,73,9,6,40,72,204,245,...,,3,7,4,26,42,66,99,43,20,273,364,476,62,776,...,,4,0,22,42,80,32,27,335,504,728,038,428,944,...,,4,2,30,66,32,246,429,75,44,768,2652,3876,...,,5,5,43,99,27,429,80,430,2438,3978,630,...,,5,9,55,43,335,75,430,2704,4862,8398,...,,6,22,73,20,504,44,2438,4862,9252, The OEIS [5] now contains three versions of this table: see entries A047996, A24926, an A The table appears to be symmetric about the main iagonal, although this is not obvious from the efinition. For example, the entry T (3, 5 = 7 correspons to these seven u-vectors: 00000, 0002, 00, 00222, 022, 2, ( These vectors have five components, entries 0,, an 2, an a sum which is a multiple of 3. On the other han, the entry T (5, 3 = 7 correspons to these seven u-vectors: 000, 04, 023, 3, 22, 244, 334. (2

2 These have three components, entries from 0 to 4, an a sum which is a multiple of 5. There is no obvious bijection between the two lists. Consulting the OEIS, it appears that the nth row of the table gives (a the coefficients of the Molien series for the regular representation of the cyclic group of orer n, an also (b the numbers of inequivalent neclaces with n + beas, n of one color an of a another color. (Rows 2, 3,... appear to match OEIS entries A00869, A007997, A00860, A008646, A0329, A03292, A03293, etc. It is the purpose of this note to use Molien s theorem to prove that these empirical observations are in fact correct. 2 Molien s theorem Molien s theorem states that if G is a finite group of complex n n matrices, an a ( = 0,,... is the number of linearly inepenent homogeneous polynomials of egree that are invariant uner the action of G, then Φ G (λ = =0 a λ = G M G et(i λm (see for example [4]; [2, p. 77], [7, p. 87], [8, p. 29]. Φ G (λ is calle the Molien series for G. We apply the theorem to two ifferent (but equivalent matrix groups, G an G 2, both abstractly isomorphic to the cyclic group C n of orer n. The first, G, is the group of n n permutation matrices generate by the cyclic permutation σ := (0,, 2,..., n. For example, if n = 4, G consists of the four powers of the matrix ( The secon group, G 2, is the group of n n iagonal matrices generate by the matrix (3 iag{, ζ, ζ 2,..., ζ n }, (5 where ζ = e 2πi/n. For n = 4, the generator is iag{, i,, i}. Since G 2 is in fact the result of iagonalizing the elements of G, corresponing elements of the two groups have the same eigenvalues, an G an G 2 have the same Molien series. We will show that the invariants of G of egree correspon to neclaces of n+ beas, n of one color an of another color, an that the invariants of G 2 of egree correspon to ways to write 0 as a sum of terms in Z/nZ. Evaluating the Molien series for G gives a familiar formula (see (8 for the number of such neclaces, an since the two Molien series are the same, this is also the number of moular partitions T (n,. The results are summarize in Theorem. The fact that T (n, = T (, n is then an immeiate corollary. 3 The group G an its invariants The group G is the regular permutation representation of the cyclic group C n. It is wellnown that for each iviing n, C n contains ϕ( elements which are a prouct of n/ 2

3 cycles of length, where ϕ( is the Euler totient function (A00000 (see for example []. Each such element contributes a term ( λ n/ to the Molien series, so Φ G (λ = ϕ(. (6 n ( λ n/ n We wish to etermine a, the coefficient of λ in this expression. The contribution to the sum from the term will be zero unless also ivies, so we may restrict the sum to values of that ivie the greatest common ivisor of n an. Using the binomial theorem, we obtain a = n gc(n, This may be rewritten more symmetrically as a := n + ( n ϕ( + gc(n, ϕ(. (7 ( n+, (8 an now it is obvious that interchanging the roles of n an leaves the number unchange (as it must, since we can obviously exchange the names of the colors of the two ins of beas without affecting the number of neclaces. This is a classical formula (Lucas, [3, pp ], Rioran, [6, p. 62]. It can also be easily erive using Pólya s theorem (cf. []. Next we consier the invariants themselves an show how they correspon to neclaces. Let the variables on which G acts be labele x 0, x,..., x n. If we tae an arbitrary monomial of egree, say x l 0 0 x l x l n n, with j l j =, (9 an form the sum of its images uner all cyclic shifts of the subscripts, the result is invariant uner the action of G, an conversely, every invariant can be ecompose into a sum of such invariants. We escribe neclaces with n + beas of two colors by binary vectors of length n +, containing n 0 s an s, with the unerstaning that cyclic shifts of the vector correspon to the same neclace. For example, there are five neclaces with four beas of one color an three of the other color, which are escribe by the binary vectors 0000, 0000, 0000, 0000, (0 The corresponence between invariants of egree an neclaces with n + beas is that a monomial (9 correspons to the neclace with binary vector 0 l 0 0 l 0 l 2 0 l n, ( where there are n 0 s an s. In other wors, the exponents in the monomial (9 specify the lengths of the successive strings of beas of one color. Choosing a ifferent monomial term from an invariant just gives a ifferent cyclic shift of the neclace. The invariants corresponing to the neclaces in (0 are respectively x 4 0+x 4 +x 4 2, x 3 0x +x 3 x 2 +x 3 2x 0, x 3 0x 2 +x 3 x 0 +x 3 2x, x 2 0x 2 +x 2 x 2 2+x 2 2x 2 0, x 2 0x x 2 +x 2 x 2 x 0 +x 2 2x 0 x. 3

4 4 The group G 2 an its invariants The eigenvalues of the rth power of the generator (5 are ζ jr, j = 0,,..., n, so the Molien series for G 2 is Φ G2 = n n n j=0 ( λζjr. (2 r=0 Since G 2 consists of iagonal matrices, any monomial term in an invariant must itself be invariant. If (9 is an invariant of egree, then we have n l j =, j=0 n jl j 0 mo n, (3 j=0 which is the efinition of a moular partition of n into parts, the parts u, u 2,..., u being 0 (l 0 times, (l times,..., n (l n times. So by Molien s theorem, Φ G2 (λ = T (n, λ. =0 Since Φ G (λ = Φ G2 (λ, we have prove: Theorem The following are equal: (i T (n,, the number of ways of writing 0 as a sum of terms in Z/nZ, (ii the number of bi-colore neclaces with n beas of one color an beas of another color, (iii Corollary 2 T (n, = T (, n. n + gc(n, ϕ( ( n+. (4 References [] N. G. e Bruijn, Polyà s theory of counting, in E. F. Becenbach, e., Applie Combinatorial Mathematics, Wiley, 964, pp ; available from [2] H. Dersen an G. Kemper, Computational Invariant Theory, Springer, [3] E. Lucas, Théorie es Nombres, Gauthier-Villars, Paris, Vol., 89. [4] T. Molien, Über ie Invarianten er linear Substitutions-gruppe, Sitzungsber. König. Preuss Aa. Wiss., (897, One can also show irectly, by purely combinatorial arguments, that the right-han sie of (2 is a generating function for the numbers T (n, for fixe n. 4

5 [5] The OEIS Founation, Inc., The On-Line Encyclopeia of Integer Sequences, publishe electronically at [6] J. Rioran, An Introuction to Combinatorial Analysis, Wiley, NY, 958. [7] L. Smith, Polynomial Invariants of Finite Groups, Peters, Wellesley, MA, 995. [8] B. Sturmfels, Algorithms in Invariant Theory, Springer, 993. [9] J. Voß, Posting to Sequence Fans Mailing List, April 30,