A011979 -id:A011979 - cp's OEIS frontend

A066010 Triangle of covering numbers T(n,k) = C(n,k,k-1), n >= 2, 2 <= k <= n.

1, 2, 1, 2, 3, 1, 3, 4, 4, 1, 3, 6, 6, 5, 1, 4, 7, 12, 9, 6, 1, 4, 11, 14, 20, 12, 7, 1, 5, 12, 25, 30, 30, 16, 8, 1, 5, 17, 30, 51, 50, 45, 20, 9, 1, 6, 19, 47, 66

Offset: 2

N. J. A. Sloane, Dec 30 2001

C(v,k,t) is the smallest number of k-subsets of an n-set such that every t-subset is contained in at least one of the k-subsets.

			Table of values of C(v,k,k-1):
v\k.2..3..4...5...6...7...8..9.10.11.12.13
.2 .1
.3 .2..1
.4 .2..3..1
.5 .3..4..4...1
.6 .3..6..6...5...1
.7 .4..7.12...9...6...1
.8 .4.11.14..20..12...7...1
.9 .5.12.25..30..30..16...8..1
10 .5.17.30..51..50..45..20..9..1
11 .6.19.47..66...a..84..63.25.10..1
12 .6.24.57.113.132...b.126.84.30.11..1
13 .7.26.78.???.245.???..?.185.??.36.12.1
where a in range 96-100, b in range 165-176

CRC Handbook of Combinatorial Designs, 1996, p. 263.
W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992.

D. Applegate, E. M. Rains and N. J. A. Sloane, On asymmetric coverings and covering numbers, J. Comb. Des. 11 (2003), 218-228.
D. Gordon, La Jolla Repository of Coverings
K. J. Nurmela and Patric R. J. Östergård, New coverings of t-sets with (t+1)-sets, J. Combinat. Designs, 7 (1999), 217-226.
K. J. Nurmela and Patric R. J. Östergård, New coverings of t-sets with (t+1)-sets (appendix), J. Combinat. Designs, 7 (1999), 217-226.
Index entries for covering numbers

Columns give A011975, A011979, A011983, A011987, A066009, A066011, A066137, A066140, A066225.
Triangle in A066701 gives number of nonisomorphic solutions.
Triangle in A036838 (the Schoenheim bound) gives lower bounds to these entries.

A036831 Schoenheim bound L_1(n,4,3).

Original entry on oeis.org

1, 4, 6, 11, 14, 25, 30, 47, 57, 78, 91, 124, 140, 183, 207, 257, 285, 352, 385, 466, 510, 600, 650, 763, 819, 950, 1020, 1163, 1240, 1411, 1496, 1689, 1791, 1998, 2109, 2350, 2470, 2737, 2877, 3161, 3311, 3634, 3795, 4148, 4332, 4704, 4900, 5317, 5525, 5976

Offset: 4

N. J. A. Sloane, Jan 11 2002

nonn

W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992. See Eq. 1.

Index entries for covering numbers

Lower bound to A011979. Cf. A011975.

A column of A036838.

L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Current sequence is L_1(n,4,3,1).

L[v_, k_, t_, l_] := Module[{i, t1}, t1 = l; For[i = v - t + 1, i <= v, i++, t1 = Ceiling[t1*i/(i - (v - k))]]; t1];
T[n_, k_] := L[n + 2, k + 2, k + 1, 1];
a[n_] := T[n - 2, 2];
Table[a[n], {n, 4, 49}] (* Jean-François Alcover, Mar 07 2023, after Maple code *)

cp's OEIS Frontend

A066010 Triangle of covering numbers T(n,k) = C(n,k,k-1), n >= 2, 2 <= k <= n.

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