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.

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

0,4

Comments

Let x be in [n]. Then x is accessible by the binary relation R if (x,x) is in R^j for some j>=1. In other words, x is accesible by R if (x,x) is in the transitive closure of R. See Schwarz link.

Links

Table of n, a(n) for n=0..27.

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.

S. Schwarz, On the semigroup of binary relations on a finite set, Czechoslovak Mathematical Journal, 1970.

Formula

Sum_{n>=0} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(s(2yx)-yx + x))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)), s(x) = Sum_{n>=0} A003030(n)x^n/n! and @ is the exponential Hadamard product (see Panafieu and Dovgal).

Example

Triangle begins:
     1;
     1,      1;
     3,      6,      7;
    25,     75,    159,     253;
   543,   2172,   6354,   17004,   39463;
 29281, 146405, 532130, 1841650, 6808765, 24196201;
...

Mathematica

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}]]

Crossrefs

Cf. A002416 (row sums), A003024 (column k = 0), A366866 (main diagonal), A003030.

Sequence in context: A288804 A288335 A362958 * A288653 A287905 A287947

Adjacent sequences: A370200 A370201 A370202 * A370204 A370205 A370206

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 11 2024

Status

approved