A370385
Triangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent relation in {R^i:i>=1} is a quasi-order containing exactly k strongly connected components.
Original entry on oeis.org
1, 1, 3, 4, 139, 66, 48, 25575, 9280, 3072, 1536, 18077431, 4498530, 1174800, 322560, 122880
Offset: 1
Triangle begins:
1;
1;
3, 4;
139, 66, 48;
25575, 9280, 3072, 1536;
18077431, 4498530, 1174800, 322560, 122880;
...
-
nn = 5; B[n_] := n! 2^Binomial[n, 2]; s[x_, y_] := y x + (3 y + y^2) x^2/2! + (139 y + 3 y^2 + 2 y^3) x^3/3! + (25575 y + 103 y^2 + 12 y^3 + 6 y^4) x^4/
4! + (18077431 y + 4815 y^2 + 230 y^3 + 60 y^4 + 24 y^5) x^5/5! ;
ggf[egf_] :=Normal[Series[egf, {x, 0, nn}]] /.Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[
Series[1/ggf[Exp[-s[x, y]]], {x, 0, nn}], {x, y}]]
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
- 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]
A370203
Triangular array read by rows. T(n,k) is the number of binary relations on [n] that have exactly k accessible points, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 3, 6, 7, 25, 75, 159, 253, 543, 2172, 6354, 17004, 39463, 29281, 146405, 532130, 1841650, 6808765, 24196201, 3781503, 22689018, 97165485, 395729820, 1801073385, 9917482698, 56481554827
Offset: 0
Triangle begins:
1;
1, 1;
3, 6, 7;
25, 75, 159, 253;
543, 2172, 6354, 17004, 39463;
29281, 146405, 532130, 1841650, 6808765, 24196201;
...
-
nn = 6; B[n_] := n! 2^Binomial[n, 2]; 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}]];
ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(s[2 y x] - y x + x)]], {x, 0, nn}], {x, y}]]
Showing 1-3 of 3 results.
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