A328773 Irregular triangle read by rows: T(n,k) is the number of colored digraphs on n nodes with color scheme given by the partitions of n in canonical ordering.

1, 1, 3, 4, 16, 36, 64, 218, 752, 1104, 2112, 4096, 9608, 45960, 90416, 178944, 266496, 528384, 1048576, 1540944, 9133760, 22692704, 45277312, 30194176, 90196736, 180011008, 135032832, 269500416, 537919488, 1073741824

Offset: 0

Author

Peter Dolland, Oct 27 2019

Keywords

• nonn
• tabf

Comments

Colors are not interchangeable. Adjacent nodes may have the same color.

A partition [b_1, ..., b_m] with b_i > 0 and Sum_{i=1..m} b_i = n corresponds to a color scheme on n nodes having m colors. To find out which digraphs are equivalent with respect to a color scheme, consider the automorphism group on the partition. This group is the m-fold product of the symmetric groups on the b_i nodes, and therefore contains Product_{i=1..m} b_i! elements (i.e. the permutations).

Calculate the number of equivalence classes by determining the cycle index of the group induced by the automorphism group in the set of the edges [(i,j)|i,j in [1..n]; i != j] and set, with Pólya, the variable values to 2.

The left column of the triangle gives the number of unlabeled digraphs, while the right flank of the triangle gives the number of labeled digraphs.

Canonical ordering is also known as graded reverse lexicographic ordering, see A080577, A063008, or link below. Partitions here have the property b_i >= b_j for i < j.

Examples

			The sequence begins:
      1;
      1;
      3,       4;
     16,      36,       64;
    218,     752,     1104,     2112,     4096;
   9608,   45960,    90416,   178944,   266496,   528384,   1048576;
   ...
For n = 3, the three partitions of n are [3], [2, 1] and [1, 1, 1]. T(n,1) = 16 gives the number of colored digraphs with all nodes having the same color; T(n, 2) = 36 gives the number of colored digraphs with two nodes having the first color and one node having the second color; T(n, 3) gives the number of colored digraphs with each node having its own color.
For n = 5, k = 4 the required partition is [3,1,1]. T(5,4) = 178944 is then the number of colored digraphs with 5 nodes, where 3 nodes have the first color and the other two nodes each has its own color.
		

References

• N. G. de Bruijn, Pólyas Abzähl-Theorie: Muster für Graphen und chemische Verbindungen, Selecta Mathematica III, Springer-Verlag (1971), 1-55.

Links

• Peter Dolland, Table of n, a(n) for n = 0..138 (rows 0..10)
• OEIS Wiki, Orderings of partitions (a comparison).

Crossrefs

Cf. A000041 equals the row length, A080577 lists the partitions in the used order, A063008 instantiates the index sequences encoding the partitions. A000273 and A053763 represent the flanks of the triangle.

Programs

PARI

\\ here C(p) computes sequence value for given partition.
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i]-1)}
C(p)={((i,v)->if(i>#p, 2^edges(v), my(s=0); forpart(q=p[i], s+=permcount(q)*self()(i+1, concat(v,Vec(q)))); s/p[i]!))(1, [])}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], ,4))}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Nov 02 2019

Formula

T(n, 1) = A000273(n).

T(n, A000041(n)) = A053763(n) = 2^(n^2 - n).

T(n, A000041(n)-1) = 2^(n^2 - 3*n - 1) * (2^(2*n) + 8) for n > 1.