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A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n. 1
1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

All the enumerated lattices have full rank k, since the quotient group is finite.

For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.

Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].

T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).

Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.

Columns are multiplicative functions.

REFERENCES

L. M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., vol. 112, no. 539, 1994.

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]

LINKS

Álvar Ibeas, First 100 antidiagonals, flattened

J. H. Kwak, J.-H. Chun, and J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]

FORMULA

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).

T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).

Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).

If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).

EXAMPLE

There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.

Array begins:

      k=1    k=2    k=3    k=4    k=5    k=6

n=1     1      1      1      1      1      1

n=2     1      3      7     15     31     63

n=3     1      4     13     40    121    364

n=4     1      6     28    120    496   2016

n=5     1      6     31    156    781   3906

n=6     1     12     91    600   3751  22932

CROSSREFS

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).

Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.

Cf. A160870.

Sequence in context: A058879 A208344 A209172 * A160870 A025255 A296006

Adjacent sequences:  A263947 A263948 A263949 * A263951 A263952 A263953

KEYWORD

nonn,tabl

AUTHOR

Álvar Ibeas, Oct 30 2015

STATUS

approved

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Last modified May 23 04:59 EDT 2019. Contains 323508 sequences. (Running on oeis4.)