A109452 Maximum of min(primeimplicants(f),primeimplicants(NOT f)) over all symmetric Boolean functions of n variables.

1, 1, 4, 5, 21, 31, 113, 177, 766, 1271, 4687, 7999, 34412, 60166, 225891, 401201, 1702653, 3064183, 11646431, 21171246, 88894429, 162966750, 624746839, 1153324813, 4805206256, 8923870307, 34421146489, 64252106507, 266183327326

Offset: 1

Author

Don Knuth, Aug 27 2005

Keywords

• nonn
• easy

Comments

Fridshal's example for n=9 was S_{2,3,4,8,9}(x_1,...,x_9); this has "only" 765 prime implicants.

Examples

			a(9)=766 because of the symmetric function S_{0,2,3,4,8}(x_1, ..., x_9).
		

References

• R. Fridshal, Summaries, Summer Institute for Symbolic Logic, Department of Mathematics, Cornell University, 1957, 211-212.
• D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1 (in preparation).

Crossrefs

Cf. A109385, A109388.

Programs

Maple

aux := proc(m,n) option remember ; if m < 0 then 0 ; else combinat[multinomial](n,ceil(m/2),floor(m/2),n-m)+binomial(n,ceil(m/2-1))+aux(ceil(m/2)-2,n) ; fi ; end: A109452 := proc(n) aux( ceil(n/2)-1,n) ; end: for n from 1 to 40 do printf("%d, ",A109452(n)) ; od ; # R. J. Mathar, May 08 2007

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
aux[m_, n_] := aux[m, n] = If [m<0, 0, multinomial[n, {Ceiling[m/2], Floor[m/2], n-m}]+Binomial[n, Ceiling[m/2-1]]+aux[Ceiling[m/2]-2, n]];
a[n_] := aux[Ceiling[n/2]-1, n];
Array[a, 40] (* Jean-François Alcover, Mar 24 2021, after R. J. Mathar *)

Formula

a(n) = aux(ceiling(n/2) - 1, n), where aux(m, n) = trinomial(n, ceiling(m/2), floor(m/2), n-m) + binomial(n, ceiling(m/2-1)) + aux(ceiling(m/2)-2, n).

Extensions

More terms from R. J. Mathar, May 08 2007