A306419 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.

1, 1, 1, 1, 4, 11, 32, 99, 326, 1123, 4064, 15291, 59924, 242945, 1019584, 4409233, 19648674, 89938705, 422744384, 2035739041, 10039057524, 50610247483, 260704414816, 1370387233859, 7346982653702, 40131663286851, 223238920709024, 1263531826402891, 7273434344119460

Offset

0,5

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Formula

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

Example

The a(1) = 1 through a(5) = 11 set partitions:
  {{1}}  {{1}{2}}  {{1}{2}{3}}  {{13}{24}}      {{1}{24}{35}}
                                {{1}{24}{3}}    {{13}{24}{5}}
                                {{13}{2}{4}}    {{13}{25}{4}}
                                {{1}{2}{3}{4}}  {{14}{2}{35}}
                                                {{14}{25}{3}}
                                                {{1}{2}{35}{4}}
                                                {{1}{24}{3}{5}}
                                                {{1}{25}{3}{4}}
                                                {{13}{2}{4}{5}}
                                                {{14}{2}{3}{5}}
                                                {{1}{2}{3}{4}{5}}

Mathematica

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

Prog

(PARI) \\ here b(n) is A000085(n)
b(n) = {sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
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

Crossrefs

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

Sequence in context: A199109 A025268 A178520 * A376071 A149232 A149233

Adjacent sequences: A306416 A306417 A306418 * A306420 A306421 A306422

Keyword

nonn

Author

Gus Wiseman, Feb 14 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 30 2019

Status

approved