A365593 -id:A365593 - 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).

A366141 Triangular array read by rows: T(n,k) is the number of Boolean relation matrices such that all of the blocks of its Frobenius normal form are 0-blocks or 1-blocks and that have exactly k 1-blocks on the diagonal, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 7, 3, 25, 85, 84, 25, 543, 2335, 3579, 2322, 543, 29281, 152101, 310020, 309725, 151835, 29281, 3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503, 1138779265, 8051910805, 24318772884, 40667112045, 40664902810, 24315521720, 8050866418, 1138779265

Offset: 0

Geoffrey Critzer, Sep 30 2023

A 1(0) block is such that every entry in the block is 1(0).

Conjecture: lim_{n -> oo} T(n,k)/T(n,n-k) = 1.

			Triangle begins ...
        1;
        1,        1;
        3,        7,        3;
       25,       85,       84,       25;
      543,     2335,     3579,     2322,      543;
    29281,   152101,   310020,   309725,   151835,    29281;
  3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503;
  ...

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. A365593 (row sums), A003024.

nn = 6; B[n_] := 2^Binomial[n, 2] n!; dags=Select[Import["https://oeis.org/A003024/b003024.txt", "Table"],
   Length@# == 2 &][[All, 2]]; d[x_] := Total[dags Table[x^i/i!, {i, 0, 40}]];
Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[
    Series[d[y (Exp[x] - 1) + x], {x, 0, nn}], {x, y}]] // Grid

T(n,0) = T(n,n) = A003024(n).

E.g.f.: D(y(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.

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A366141 Triangular array read by rows: T(n,k) is the number of Boolean relation matrices such that all of the blocks of its Frobenius normal form are 0-blocks or 1-blocks and that have exactly k 1-blocks on the diagonal, n>=0, 0<=k<=n.

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

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