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%I #7 Jan 21 2023 02:28:32
%S 1,3,6,9,15,19,26,36,46,59,80,100,128,167,211,267,341,429,541,682,850,
%T 1063,1327,1647,2035,2520,3100,3810,4669,5708,6955,8468,10267,12441,
%U 15026,18120,21788,26175,31355,37510,44769,53362,63460,75384,89348
%N 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 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 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 For m >= 3, a(2m) = A000041(m) + A001399(m-3) + A000005(m+1) + A083039(m-2) + m - 5.
%e 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 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 Cf. A000041, A001399, A000005, A083039.
%K nonn
%O 1,2
%A _Michael David Hirschhorn_ and _James A. Sellers_, Sep 06 2008