A001929 -id:A001929 - cp's OEIS frontend

A046910 Number of connected irreducible quasiorders with n labeled points.

1, 1, 1, 1, 25, 1321, 70201, 4542721, 384969649, 44087846545, 6926924885881, 1503058888234201, 451117640363382697, 186980881340749198561, 106678398214255092939169, 83440038893764124092029601, 89093417035281194970121062073, 129323858612953057624127147727913, 254190262374139251098507525465587609

Offset: 0

John A. Wright

nonn

John A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
John A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A046909, A335987, A000798, A001929.

nn = 18; A[x_] :=Total[Cases[Import["https://oeis.org/A000798/b000798.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[Series[1 + Log[A[x]] - A[x] (1 - 1/A[x])^2, {x, 0, nn}], x] (* Geoffrey Critzer, Jul 10 2022 *)

From Geoffrey Critzer, Jul 10 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)*(1 - 1/A(x))^2 where A(x) is the e.g.f. for A000798.
a(n) = A001929(n) - Sum_{k>=2} A335987(n,k). (End)

a(8)-a(18) from Geoffrey Critzer, Jul 10 2022

A247232 Triangular array read by rows: T(n,k) is the number of pre-orders on an n-set with exactly k connected components in its digraph representation, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 3, 1, 19, 9, 1, 233, 103, 18, 1, 4851, 1735, 325, 30, 1, 158175, 43201, 7320, 785, 45, 1, 7724333, 1567783, 218491, 22960, 1610, 63, 1, 550898367, 82142943, 8856974, 818461, 59570, 2954, 84, 1, 56536880923, 6187176225, 496368181, 37205658, 2518131, 135198, 4998, 108, 1

Offset: 1

Geoffrey Critzer, Nov 27 2014

The Bell transform of A001929(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			1;
3,         1;
19,        9,        1;
233,       103,      18,      1;
4851,      1735,     325,     30,     1;
158175,    43201,    7320,    785,    45,    1;
7724333,   1567783,  218491,  22960,  1610,  63,   1;
550898367, 82142943, 8856974, 818461, 59570, 2954, 84, 1;

Column 1 = A001929.

Row sums = A000798.

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
Rest[CoefficientList[#, y]]& /@ (CoefficientList[A[Exp[x] - 1]^y + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* Jean-François Alcover, Jan 01 2020 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
topo = oeis('A001929')  # Fetch the data via Internet.
A001929List = topo.first_terms()
A001929 = lambda n: A001929List[n]
bell_matrix(lambda n: A001929(n+1), 10) # Peter Luschny, Jan 18 2016, updated Mar 25 2020

E.g.f.: A(exp(x)-1)^y where A(x) is the e.g.f. for A001035.

cp's OEIS Frontend

A046910 Number of connected irreducible quasiorders with n labeled points.

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