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

1,5

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.

Example

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, ...

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

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 *)

Crossrefs

Rows give A003319, A027837, A049291.

Columns give A000225, A049294, A049295.

Main diagonal is A057014.

Sequence in context: A112492 A210574 A353532 * A297191 A147990 A134567

Adjacent sequences: A049287 A049288 A049289 * A049291 A049292 A049293

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Sep 09 2000

Extensions

More terms from Alois P. Heinz, Oct 29 2009

Status

approved