A049290 Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.
1, 1, 1, 1, 3, 1, 1, 7, 13, 1, 1, 15, 97, 71, 1, 1, 31, 625, 2143, 461, 1, 1, 63, 3841, 54335, 68641, 3447, 1, 1, 127, 23233, 1321471, 8563601, 3011263, 29093, 1, 1, 255, 139777, 31817471, 1035045121, 2228419359, 173773153, 273343, 1, 1, 511, 839425
Offset: 1
Examples
Array T(n,k) (n >= 1, k >= 1) begins: 1, 1, 1, 1, 1, ... 1, 3, 13, 71, 461, ... 1, 7, 97, 2143, 68641, ... 1, 15, 625, 54335, 8563601, ...
References
- P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
- 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.
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
Links
- Alois P. Heinz, Antidiagonals n = 1..37, flattened
- J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3. [Broken link?]
- V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
Crossrefs
Programs
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Maple
T:= proc(n,k) option remember; k* k!^(n-1) -add(j!^(n-1) *T(n, k-j), j=1..k-1) end: seq(seq(T(d+1-k, k), k=1..d), d=1..10); # Alois P. Heinz, Oct 29 2009
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Mathematica
nmax = 10; t[n_, k_] := t[n, k] = k*k!^(n-1) - Sum[j!^(n-1)*t[n, k-j], {j, 1, k-1}]; Flatten[ Table[ t[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 09 2011, after Maple *)
Extensions
More terms from Alois P. Heinz, Oct 29 2009