A133721 Triangle read by rows: T(m,n) = number of n-balanced and minimal labeled covers of a finite set of m unlabeled elements (m >= 1, 1 <= n <= m).
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 7, 1, 1, 1, 1, 1, 1, 1, 15, 6, 3, 22, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 21, 65, 81, 7, 34, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10
Offset: 1
Examples
Triangle begins: 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 6 7 1 1 1 1 1 1 3 1 1 1 1 1 1 10 1 13 1 1 1 1 1 1 1 25 7 1 1 1 1 1 1 1 15 6 3 22 1 1 1 1 1 1
Links
- A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.
Crossrefs
Programs
-
Maple
A133721 := proc(m,n) l := ceil(m/n) ; c := n*ceil(m/n)-m ; A133713(l,c) ; end proc: # R. J. Mathar, Nov 23 2011
-
Mathematica
A133713[l_, cl_] := Module[{g, k, s}, g = 1; For[k = 1, k <= cl+1, k++, s = Sum[Binomial[Binomial[l, k+1] + i-1, i]*t^(i*k), {i, 0, Ceiling[cl/k]}]; g = g*s]; g = Expand[g]; SeriesCoefficient[g, {t, 0, cl}]]; A133713[, 0] = 1; a[m, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m]; Table[a[m, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)