1,2

S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506.

S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147.

For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.

For m = 4, the number of unigraphical partitions is A000041(4) + A001399(1) + A000005(5) + A083039(2) + 4 - 5 = 5 + 1 + 2 + 2 + 4 - 5 = 9.

with(combinat): with(numtheory): a:=proc(m) it:=round(m^2/12)+numbpart(m)+tau(m+1)+m-5: if m mod 6 = 0 then it:=it+2 fi: if m mod 6 = 1 then it:=it+1 fi: if m mod 6 = 2 then it:=it+3 fi: if m mod 6 = 3 then it:=it+1 fi: if m mod 6 = 4 then it:=it+2 fi: if m mod 6 = 5 then it:=it+2 fi: RETURN(it): end:

Cf. A000041, A001399, A000005, A083039.

Sequence in context: A355498 A049991 A368611 * A031940 A007187 A337502

Adjacent sequences: A143978 A143979 A143980 * A143982 A143983 A143984

Michael David Hirschhorn and James Sellers, Sep 06 2008

approved