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A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).

%I A160449 #24 Jan 28 2025 01:53:20
%S A160449 1,1,1,1,2,1,1,3,4,1,1,5,11,8,1,1,7,43,49,16,1,1,11,161,681,251,32,1,
%T A160449 1,15,901,14721,14491,1393,64,1,1,22,5579,524137,1730861,336465,8051,
%U A160449 128,1,1,30,43206,25471105,373486525,207388305,7997683,47449,256,1
%N A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).
%C A160449 T(n,k) is the number of orbits of the conjugacy action of Sym(n) on Sym(n)^k [Kwak and Lee, 2001]. - _Álvar Ibeas_, Mar 25 2015
%H A160449 Álvar Ibeas, <a href="/A160449/b160449.txt">Table of n, a(n) for n = 0..1829</a>
%H A160449 Michael W. Hero and Jeb F. Willenbring, <a href="https://doi.org/10.1016/j.disc.2009.06.021">Stable Hilbert series as related to the measurement of quantum entanglement</a>, Discrete Mathematics, 309 (2009), 6508-6514. See Table 3.
%H A160449 J. H. Kwak and J. Lee, <a href="http://dx.doi.org/10.4153/CJM-1990-039-3">Isomorphism classes of graph bundles</a>. Can. J. Math., 42(4), 1990, pp. 747-761.
%H A160449 J. H. Kwak and J. 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. See Table 2.
%e A160449 The array begins:
%e A160449       k=0   k=1   k=2   k=3     k=4       k=5
%e A160449   n=1   1     1     1     1       1         1
%e A160449   n=2   1     2     4     8      16        32
%e A160449   n=3   1     3    11    49     251      1393
%e A160449   n=4   1     5    43   681   14491    336465
%e A160449   n=5   1     7   161 14721 1730861 207388305
%o A160449 (Sage)
%o A160449 def A160449(n, k):
%o A160449     return sum(p.aut()**(k - 1) for p in Partitions(n))
%o A160449 # _Álvar Ibeas_, Mar 25 2015
%Y A160449 Rows: A000012, A000079, A074528, A160450, A160454, A176709.
%Y A160449 Columns: A000041, A110143, A152612, A160446, A160447, A160448.
%Y A160449 Cf. A057004.
%K A160449 nonn,tabl
%O A160449 0,5
%A A160449 _N. J. A. Sloane_, Nov 13 2009
%E A160449 Name clarified and more terms added by _Álvar Ibeas_, Mar 25 2015