A306419 - OEIS

%I #9 Aug 30 2019 21:46:36

%S 1,1,1,1,4,11,32,99,326,1123,4064,15291,59924,242945,1019584,4409233,

%T 19648674,89938705,422744384,2035739041,10039057524,50610247483,

%U 260704414816,1370387233859,7346982653702,40131663286851,223238920709024,1263531826402891,7273434344119460

%N Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.

%C Also the number of spanning subgraphs of the complement of an n-cycle, with no overlapping edges.

%H Andrew Howroyd, <a href="/A306419/b306419.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*A034807(n, k)*A000085(n-2*k) for n > 2. - _Andrew Howroyd_, Aug 30 2019

%e The a(1) = 1 through a(5) = 11 set partitions:

%e   {{1}}  {{1}{2}}  {{1}{2}{3}}  {{13}{24}}      {{1}{24}{35}}

%e                                 {{1}{24}{3}}    {{13}{24}{5}}

%e                                 {{13}{2}{4}}    {{13}{25}{4}}

%e                                 {{1}{2}{3}{4}}  {{14}{2}{35}}

%e                                                 {{14}{25}{3}}

%e                                                 {{1}{2}{35}{4}}

%e                                                 {{1}{24}{3}{5}}

%e                                                 {{1}{25}{3}{4}}

%e                                                 {{13}{2}{4}{5}}

%e                                                 {{14}{2}{3}{5}}

%e                                                 {{1}{2}{3}{4}{5}}

%t stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];

%t Table[Length[stableSets[Complement[Subsets[Range[n],{2}],Sort/@Partition[Range[n],2,1,1]],Intersection[#1,#2]!={}&]],{n,0,10}]

%o (PARI) \\ here b(n) is A000085(n)

%o b(n) = {sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}

%o a(n) = {if(n < 3, n >= 0, sum(k=0, n\2, (-1)^k*b(n-2*k)*n*(n-1-k)!/(k!*(n-2*k)!)))} \\ _Andrew Howroyd_, Aug 30 2019

%Y Cf. A000085, A000110, A000296, A001006, A001610, A003436 (no singletons), A034807, A170941 (linear case), A278990 (linear case with no singletons), A306417.

%K nonn

%O 0,5

%A _Gus Wiseman_, Feb 14 2019

%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 30 2019