A329546 Triangle read by rows: T(n,k) is the number of colored digraphs on n nodes with exactly k colors arbitrarily assigned (1 <= k <= n).

1, 3, 4, 16, 72, 64, 218, 2608, 6336, 4096, 9608, 272752, 1336320, 2113536, 1048576, 1540944, 93847104, 812045184, 2337046528, 2689597440, 1073741824, 882033440, 110518842048, 1580861402112, 7344135176192, 14676310097920, 13200581984256, 4398046511104

Offset: 1

Author

Peter Dolland, Nov 16 2019

Keywords

• nonn
• tabl

Comments

The values are weighted subtotals of the rows of the irregular triangle A328773.

The weight of a color scheme is the multiplicity A072811(n,k) with k as the index of the induced partition.

T(n,k) gives the number of digraphs (see A000273) without restrictions, where nodes of the same color are not differentiated.

If we do not consider the exchange of colors with different sizes to be different digraphs, we can impose an order on the colors, which leads to A329541.

Examples

			First six rows:
      1
      3        4
     16       72        64
    218     2608      6336       4096
   9608   272752   1336320    2113536    1048576
1540944 93847104 812045184 2337046528 2689597440 1073741824
n=4, k=2: Partitions: [3,1] and [2,2] with indices 2 and 3 and multiplicities 2 and 1: T(4,2) = Sum_{i=2,3} A072811(4,i)*A328773(4,i) = 2*752 + 1104 = 2608.
n=6, k=3: Partitions: [4,1,1], [3,2,1], [2,2,2] with indexes 4, 6, 8 and multiplicities 3, 6, 1: T(6,3) = Sum_{i=4,6,8} A072811(6,i)*A328773(6,i) = 3*45277312 + 6*90196736 + 1*135032832 = 812045184.
		

Crossrefs

Cf. A000273 (digraphs with one color), A053763 (digraphs with n colors), A328773 (digraphs to a given color scheme).

Cf. A072811 (multiplicity of color schemes).

Cf. A329541 (ordered colors).

Cf. A309980 (reflexive/anti-reflexive: just two colors).

Programs

PARI

\\ here C(p) computes A328773 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, [])}
\\ here mulp(v) computes the multiplicity of the given partition. (see A072811)
mulp(v) = {my(p=(#v)!, k=1); for(i=2, #v, k=if(v[i]==v[i-1], k+1, p/=k!; 1)); p/k!}
wC(p)=mulp(p)*C(p)
Row(n)={[vecsum(apply(wC, vecsort([Vecrev(p) | p<-partitions(n),#p==m], , 4))) | m<-[1..n]]}
{ for(n=0, 10, print(Row(n))) }

Formula

T(n,1) = A000273(n) = A328773(n,1).

T(n,n) = A053763(n) = A328773(n,A000041(n)).

T(n,n-1) = (n-1)*A328773(n,A000041(n)-1).

T(n,k) = Sum_{i=1..A000041(n), A063008(n,i) encodes a partition with k elements} A072811(n,i)*A328773(n,i).