A355612 -id:A355612 - cp's OEIS frontend

A365590 Number of n X n Boolean relation matrices such that each of the diagonal blocks of its Frobenius normal form is either a 1 block or a 0 block.

1, 2, 13, 243, 11998, 1477763, 436610299, 300960642300, 474171878424571, 1680899431189662775, 13241419272545722904788, 229482664065433754849099977, 8677282817864146616211588609715, 710901968198799834001047038898570250

Offset: 0

Geoffrey Critzer, Sep 10 2023

nonn

A 1(0) block is such that every entry in the block is 1(0). See Gregory, Kirkland, Pullman for a description of Frobenius normal form.

a(n) is also the number of labeled digraphs (with loops allowed A002416) on [n] such that every strongly connected component is either complete or a single vertex without a loop.

D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.

Cf. A355612, A365593, A365534.

nn = 13; B[n_] := n! 2^Binomial[n, 2]; ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(Exp[x] - 1 + x)]], {x, 0, nn}], x]

Sum_{n>=0} a(n)*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(exp(x)-1+x))) where E(x)=Sum_{n>=0} x^n/(n!*2^binomial(n,2)) and @ is the exponential Hadamard product (see Panafieu and Dovgal).

A365593 Number of n X n Boolean relation matrices such that every block of its Frobenius normal form is either a 0 block or a 1 block.

Original entry on oeis.org

1, 2, 13, 219, 9322, 982243, 249233239, 148346645212, 202688186994599, 624913864623500599, 4289324010827093793808, 64841661094150427710360745, 2140002760057211517052090865983, 153082134018816602622335941790247946, 23590554099141037133024176892280338280237

Offset: 0

Geoffrey Critzer, Sep 10 2023

nonn

A 1(0) block is such that every entry in the block is 1(0). If a Boolean relation matrix R is limit dominating then it must be that every block of R is either a 0 block or a 1 block. See Theorem 1.2 in Gregory, Kirkland, and Pullman.

Conjecture: lim_n->inf a(n)/(A003024(n)*2^n) = 1. In other words, almost all of the relations counted by this sequence have n strongly connected components. - Geoffrey Critzer, Sep 30 2023

D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.

Cf. A003024, A355612, A365534, A365590, A366141.

nn = 12; d[x_] :=Total[Cases[Import["https://oeis.org/A003024/b003024.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 41}]];
Range[0, nn]! CoefficientList[Series[d[Exp[x] - 1 + x], {x, 0, nn}],x]

E.g.f.: D(exp(x)-1+x) where D(x) is the e.g.f. for A003024.

A369397 Number of binary relations R on [n] such that the (unique) idempotent in {R,R^2,R^3,...} is an equivalence relation.

Original entry on oeis.org

1, 1, 5, 157, 26345, 18218521, 47136254765, 451286947588597, 16264532016440908625, 2253156851039460378774961, 1219026648017155982267265596885, 2601923405098893502520360223043594957, 22040885615442635622424409144799379027505465

Offset: 0

Geoffrey Critzer, Jan 22 2024

nonn

Equivalently, a(n) is the number of binary relations R on [n] such that the Frobenius normal form has no 0-blocks on the diagonal and all off diagonal blocks are 0-blocks.

D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
S. Schwarz, On the semigroup of binary relations on a finite set , Czechoslovak Mathematical Journal, 1970.

Cf. A366866 (binary relations R on [n] such that the (unique) idempotent in {R,R^2,R^3,...} is a quasiorder), A365534, A366218, A365590, A355612, A365593, A366252, A366350, A366218.

nn = 12; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"],
   Length@# == 2 &][[All, 2]]; s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]];
Table[n!, {n, 0, nn}] CoefficientList[Series[Exp [s[2 x] - x], {x, 0, nn}], x]

E.g.f.: exp(s(2x)-x) where s(x) is the e.g.f. for A003030.

cp's OEIS Frontend

A365590 Number of n X n Boolean relation matrices such that each of the diagonal blocks of its Frobenius normal form is either a 1 block or a 0 block.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A365593 Number of n X n Boolean relation matrices such that every block of its Frobenius normal form is either a 0 block or a 1 block.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A369397 Number of binary relations R on [n] such that the (unique) idempotent in {R,R^2,R^3,...} is an equivalence relation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula