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A035348 Triangle of a(n,k) = number of k-member minimal covers of an n-set (n >= k >= 1).

%I A035348 #51 Feb 16 2025 08:32:37
%S A035348 1,1,1,1,6,1,1,25,22,1,1,90,305,65,1,1,301,3410,2540,171,1,1,966,
%T A035348 33621,77350,17066,420,1,1,3025,305382,2022951,1298346,100814,988,1,1,
%U A035348 9330,2619625,47708115,83384427,18151560,549102,2259,1
%N A035348 Triangle of a(n,k) = number of k-member minimal covers of an n-set (n >= k >= 1).
%C A035348 These are what Clarke calls "Minimal disordered k-covers of labeled n-set".
%H A035348 Alois P. Heinz, <a href="/A035348/b035348.txt">Rows n = 1..75, flattened</a>
%H A035348 R. J. Clarke, <a href="http://dx.doi.org/10.1016/0012-365X(90)90146-9">Covering a set by subsets</a>, Discrete Math., 81 (1990), 147-152.
%H A035348 T. Hearne and C. G. Wagner, <a href="http://dx.doi.org/10.1016/0012-365X(73)90141-6">Minimal covers of finite sets</a>, Discr. Math. 5 (1973), 247-251.
%H A035348 A. J. Macula, <a href="http://www.jstor.org/stable/2690571">Lewis Carroll and the enumeration of minimal covers</a>, Math. Mag., 68 (1995), 269-274.
%H A035348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalCover.html">Minimal Cover</a>
%F A035348 a(n,k) = Sum_{j >= 0} (-1)^j * binomial(k,j) * (2^k-1-j)^n. [Hearne-Wagner]
%F A035348 a(n,k) = (1/k!) * Sum_{j >= k} binomial(2^k-k-1,j-k)*j!*Stirling2(n,j). [Macula]
%F A035348 E.g.f.: Sum_{n>=0} (exp(y)-1)^n*exp(y*(2^n-n-1))*x^n/n!. - _Vladeta Jovovic_, May 08 2004
%e A035348 Triangle begins:
%e A035348   1;
%e A035348   1,    1;
%e A035348   1,    6,      1;
%e A035348   1,   25,     22,       1;
%e A035348   1,   90,    305,      65,       1,
%e A035348   1,  301,   3410,    2540,     171,      1;
%e A035348   1,  966,  33621,   77350,   17066,    420,   1;
%e A035348   1, 3025, 305382, 2022951, 1298346, 100814, 988,  1;
%e A035348   ...
%p A035348 a:= (n, k)-> add(binomial(2^k-k-1, m-k)*m!
%p A035348     *Stirling2(n, m), m=k..min(n, 2^k-1))/k!:
%p A035348 seq(seq(a(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Jul 02 2013
%t A035348 a[n_, k_] := Sum[ (-1)^i*(2^k-i-1)^n / (i!*(k-i)!), {i, 0, k}]; Flatten[ Table[ a[n, k], {n, 1, 9}, {k, 1, n}]] (* _Jean-François Alcover_, Dec 13 2011, after PARI *)
%o A035348 (PARI) {a(n, k) = sum(i=0, k, (-1)^i * binomial(k, i) * (2^k-1-i)^n) / k!} /* _Michael Somos_, Aug 05 1999 */
%Y A035348 Row sums are A046165. Cf. A049055, A003465, A002177.
%K A035348 nonn,tabl,easy,nice
%O A035348 1,5
%A A035348 _N. J. A. Sloane_
%E A035348 Entry improved by _Michael Somos_
%E A035348 Explicit formulas added by _N. J. A. Sloane_, Aug 05 2011