A364074 Array read by ascending antidiagonals: A(m, n) = Sum_{i=0..n} Sum_{d=0..n-i} binomial(n, d)*StirlingS2(n-d, i)*(m^(m-1) - 1)^(n-d-i).

1, 1, 2, 1, 2, 12, 1, 2, 67, 120, 1, 2, 628, 4355, 1424, 1, 2, 7779, 393128, 295234, 19488, 1, 2, 117652, 60497283, 247268752, 21036803, 307904, 1, 2, 2097155, 13841757800, 470668752866, 156500388128, 1625419909, 5539712, 1, 2, 43046724, 4398054899715, 1628524328796304, 3663682367243907, 100264147266880, 140823067772, 111259904

Offset

2,3

Comments

A(m, n) is the number of all ((m+1)^m)-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.

Links

Table of n, a(n) for n=2..46.

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.

Example

The array begins:
  1, 2,   12,      120,         1424,            19488, ...
  1, 2,   67,     4355,       295234,         21036803, ...
  1, 2,  628,   393128,    247268752,     156500388128, ...
  1, 2, 7779, 60497283, 470668752866, 3663682367243907, ...
  ...

Mathematica

A[m_, n_]:=Sum[Sum[Binomial[n, d]StirlingS2[n-d, i](m^(m-1)-1)^(n-d-i), {d, 0, n-i}], {i, 0, n}]; Table[A[m-n+1, n], {m, 2, 10}, {n, 0, m-2}]//Flatten

Crossrefs

Cf. A003580 (m=2), A364069 (m=3), A364070 (m=4), A364075 (antidiagonal sums).

Sequence in context: A209610 A324058 A165313 * A325636 A324121 A320834

Adjacent sequences: A364071 A364072 A364073 * A364075 A364076 A364077

Keyword

nonn,tabl

Author

Stefano Spezia, Jul 04 2023

Status

approved