|
%I #25 Nov 08 2022 08:08:11
%S 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,
%T 1,8,91,156,496,364,127,1,1,12,57,600,781,2016,1093,255,1,1,12,112,
%U 400,3751,3906,8128,3280,511,1,1,18,117,960,2801,22932,19531,32640
%N 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.
%C All the enumerated lattices have full rank k, since the quotient group is finite.
%C 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.
%C 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].
%C 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).
%C Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
%C Columns are multiplicative functions.
%D Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.
%H Álvar Ibeas, <a href="/A263950/b263950.txt">First 100 antidiagonals, flattened</a>
%H Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, <a href="http://dx.doi.org/10.1137/S0895480196304428">Enumeration of regular graph coverings having finite abelian covering transformation groups</a>, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]
%F T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
%F T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
%F Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
%F If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
%F Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
%e 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.
%e Array begins:
%e k=1 k=2 k=3 k=4 k=5 k=6
%e n=1 1 1 1 1 1 1
%e n=2 1 3 7 15 31 63
%e n=3 1 4 13 40 121 364
%e n=4 1 6 28 120 496 2016
%e n=5 1 6 31 156 781 3906
%e n=6 1 12 91 600 3751 22932
%t f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Nov 08 2022 *)
%Y Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
%Y Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
%Y Main diagonal gives A344210.
%Y Cf. A000203, A160870.
%K nonn,tabl
%O 1,5
%A _Álvar Ibeas_, Oct 30 2015
|
