A143981 The number of unigraphical partitions of 2m; that is, the number of partitions of 2m which are realizable as the degree sequence of one and only one graph (where loops are not allowed but multiple edges are allowed).

1, 3, 6, 9, 15, 19, 26, 36, 46, 59, 80, 100, 128, 167, 211, 267, 341, 429, 541, 682, 850, 1063, 1327, 1647, 2035, 2520, 3100, 3810, 4669, 5708, 6955, 8468, 10267, 12441, 15026, 18120, 21788, 26175, 31355, 37510, 44769, 53362, 63460, 75384, 89348

Offset: 1

Author

Michael David Hirschhorn and James Sellers, Sep 06 2008

Keywords

• nonn

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000041, A001399, A000005, A083039.

Programs

Maple

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:

Formula

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