A354439 Number of binary relations on [n] such that every component has at least one cycle.

1, 1, 11, 445, 62915, 33191761, 68513225711, 562467034238845, 18442237738757867675, 2417685596975700938954281, 1267626420876674359067163133991, 2658442047280176152683906485150512245, 22300713296975051923525143874710129389413715

Offset

0,3

Comments

Let A be a binary relation on [n]. Let M(A) be the unique maximal subset of [n] such that A restricted to M is nilpotent. Then a(n) is the number of relations on [n] such that M is the empty set.

Links

Table of n, a(n) for n=0..12.

Formula

E.g.f.: A(x)/B(x) where A(x) is the e.g.f. for A002416 and B(x) is the e.g.f. for A003024.

Example

a(2)=11 because all 16 binary relations on [2] have the desired property except these 5:  {{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}},
{{0, 1}, {0, 0}}, {{1, 0}, {0, 0}}.

Mathematica

nn = 12; a[p_, k_] := If[p == k, 1, Sum[(2^k - 1)^n (2^(k (p - n - k))) Binomial[p, k] a[p - k, n], {n, 1, p - k}]]; g[x_] := 1 + (Table[Sum[a[p, k], {k, 1, p}], {p, 1, nn}] Table[x^i/i!, {i, 1, nn}] // Total); h[x_] := Sum[2^(n^2) x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[h[x]/g[x], {x, 0, nn}], x]

Crossrefs

Cf. A002416, A003024.

Sequence in context: A140840 A175158 A360066 * A180087 A233219 A288685

Adjacent sequences: A354436 A354437 A354438 * A354440 A354441 A354442

Keyword

nonn

Author

Geoffrey Critzer, May 28 2022

Status

approved