cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A329874 Array read by antidiagonals: A(n,k) = number of digraphs on n unlabeled nodes, arbitrarily colored with k given colors (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 3, 3, 10, 16, 4, 21, 104, 218, 5, 36, 328, 3044, 9608, 6, 55, 752, 14814, 291968, 1540944, 7, 78, 1440, 45960, 2183400, 96928992, 882033440, 8, 105, 2456, 111010, 9133760, 1098209328, 112282908928, 1793359192848
Offset: 1

Views

Author

Peter Dolland, Nov 23 2019

Keywords

Comments

The coloring of nodes is unrestricted. There is no constraint that all of the k colors have to be used. Nodes with different colors are counted as distinct, nodes with the same color are not. For digraphs with a fixed color set see A329546.

Examples

			First six rows and columns:
      1        2          3          4           5           6
      3       10         21         36          55          78
     16      104        328        752        1440        2456
    218     3044      14814      45960      111010      228588
   9608   291968    2183400    9133760    27755016    68869824
1540944 96928992 1098209328 6154473664 23441457680 69924880288
...
n=4, k=3 with A329546:
A(4,3) = 3*218 + 3*2608 + 6336 = 14814.

Crossrefs

Cf. A000273 digraphs with one color, A000595 binary relations, A329546 digraphs with exactly k colors, A328773 digraphs with a given color scheme.

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)
A329546(n)={[vecsum(apply(wC, vecsort([Vecrev(p) | p<-partitions(n),#p==m], , 4))) | m<-[1..n]]}
Row(n)=vector(6, k, binomial(k)[2..min(k,n)+1]*A329546(n)[1..min(k,n)]~)
{ for(n=0, 6, print(Row(n))) }

Formula

A(1,k) = k.
A(2,k) = k*(2*k+1).
A(n,1) = A000273(n).
A(n,2) = A000595(n).
A(n,4) = A353996(n+1). - Brendan McKay, May 13 2022
A(n,k) = Sum_{i=1..min(n,k)} binomial(k,i)*A329546(n,i).

A383617 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, k of which are picked out, 0 <= k <= n.

Original entry on oeis.org

1, 2, 2, 10, 16, 10, 104, 272, 272, 104, 3044, 11456, 16960, 11456, 3044, 291968, 1432608, 2842304, 2842304, 1432608, 291968, 96928992, 578431232, 1441700480, 1920352256, 1441700480, 578431232, 96928992, 112282908928, 784780122880, 2351993457920, 3918054495616, 3918054495616, 2351993457920, 784780122880, 112282908928
Offset: 0

Views

Author

Peter Dolland, May 02 2025

Keywords

Comments

The row sums are the number of simple digraphs with n 4-colored nodes. The colors result from the four cases combining the property self-referencing (yes/no) with "picked out" (yes/no).

Examples

			Triangle starts:
            1;
            2,            2;
           10,           16,            10;
          104,          272,           272,           104;
         3044,        11456,         16960,         11456,          3044;
       291968,      1432608,       2842304,       2842304,       1432608,  291968;
     96928992,    578431232,    1441700480,    1920352256,    1441700480, ...
 112282908928, 784780122880, 2351993457920, 3918054495616, 3918054495616, ...
...
Example n=2, k=1: The both objects are differentiated. As a consequence all binary relations on two different objects have to be counted: These are the subsets of the cross product of the objects set with itself. This contains four pairs, so the number of subsets is 2^4 = 16.

Links

Crossrefs

Cf. A000595 (edge cases), A353996 (row sums), A329874 (4th column = row sums).

Formula

T(n,k) = T(n,n-k).
T(n,0) = T(n,n) = A000595(n).
Sum_{k=0..n} T(n,k) = A353996(n+1) = A329874(n,4).

A384105 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, exactly k of which are self referencing, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 3, 16, 36, 36, 16, 218, 752, 1104, 752, 218, 9608, 45960, 90416, 90416, 45960, 9608, 1540944, 9133760, 22692704, 30194176, 22692704, 9133760, 1540944, 882033440, 6154473664, 18425858880, 30679088480, 30679088480, 18425858880, 6154473664, 882033440
Offset: 0

Views

Author

Peter Dolland, May 19 2025

Keywords

Comments

Also the number of essentially different simple digraphs on a node set A of size n with a distinguished subset B of size k, where elements are indistinguishable within B and within A \ B.

Examples

			Triangle starts:
            1
            1,              1
            3,              4,              3
           16,             36,             36,              16
          218,            752,           1104,             752,             218
         9608,          45960,          90416,           90416,           45960, ...
      1540944,        9133760,       22692704,        30194176,        22692704, ...
    882033440,     6154473664,    18425858880,     30679088480,     30679088480, ...
1793359192848, 14334221970688, 50138592081152, 100240050239744, 125284653092864, ...
...

Links

Crossrefs

Cf. A000273 (edge cases), A000595 (row sums), A353996, A328874, A383617.

Formula

T(n,k) = T(n,n-k).
T(n,0) = T(n,n) = A000273(n).
T(n,1) = T(n,n-1) = A353996(n+1) = A329874(n,4).
Sum_{k=0..n} T(n,k) = A000595(n).
Showing 1-3 of 3 results.

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