cp's OEIS Frontend

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).

%I A143981 #8 Jul 02 2025 16:02:02
%S A143981 1,3,6,9,15,19,26,36,46,59,80,100,128,167,211,267,341,429,541,682,850,
%T A143981 1063,1327,1647,2035,2520,3100,3810,4669,5708,6955,8468,10267,12441,
%U A143981 15026,18120,21788,26175,31355,37510,44769,53362,63460,75384,89348
%N 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).
%H A143981 S. L. Hakimi, <a href="https://www.jstor.org/stable/2098746">On realizability of a set of integers as degrees of the vertices of a linear graph. I</a>, J. Soc. Indust. Appl. Math., vol. 10 (1962), 496-506.
%H A143981 S. L. Hakimi, <a href="https://www.jstor.org/stable/2098770">On realizability of a set of integers as degrees of the vertices of a linear graph. II. Uniqueness</a>, J. Soc. Indust. Appl. Math., vol. 11 (1963), 135-147.
%F A143981 For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.
%e A143981 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.
%p A143981 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:
%Y A143981 Cf. A000041, A001399, A000005, A083039.
%K A143981 nonn
%O A143981 1,2
%A A143981 _Michael David Hirschhorn_ and _James Sellers_, Sep 06 2008