A384116 -id:A384116 - cp's OEIS frontend

A303208 Number of total dominating sets in the n X n rook graph.

0, 9, 334, 53731, 30844786, 66544564805, 556588617042914, 18376877842518517955, 2414913046805958120844234, 1267171440764716263069641387581, 2658150749788131925244338204731596650, 22299981643440069703358952237798936248817875

Offset: 1

Eric W. Weisstein, Apr 19 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Total Dominating Set

Main diagonal of A384116.

Cf. A287065, A289196, A303211, A347921.

b[0] = 1; b[n_] := (2^n - 1)^n + Sum[Binomial[n, i] Sum[(-1)^j (-1 + 2^(n - j))^i Binomial[n, j], {j, 0, n}], {i, n - 1}]; Table[Sum[(-1)^k Binomial[n, k]^2 k! b[n - k], {k, 0, n}], {n, 10}]

\\ here c(n) is A287065.
b(m, n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
c(n)=(2^n-1)^n + sum(i=1, n-1, b(n, i)*binomial(n, i));
a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)^2*k!*c(n-k))} \\ Andrew Howroyd, Apr 20 2018

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^2*k!*A287065(n-k). - Andrew Howroyd, Apr 20 2018

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Apr 20 2018

Terms a(6) and beyond from Andrew Howroyd, Apr 20 2018

A384117 Array read by antidiagonals: T(n,m) is the number of minimum total dominating sets in the n X m rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 6, 3, 3, 6, 1, 1, 10, 4, 6, 4, 10, 1, 1, 15, 5, 4, 4, 5, 15, 1, 1, 21, 6, 5, 80, 5, 6, 21, 1, 1, 28, 7, 6, 65, 65, 6, 7, 28, 1, 1, 36, 8, 7, 96, 410, 96, 7, 8, 36, 1, 1, 45, 9, 8, 133, 306, 306, 133, 8, 9, 45, 1

Offset: 0

Andrew Howroyd, May 19 2025

For 1 < m <= n, the minimum size of a total dominating set is m. When 1 < m < n, solutions have exactly one vertex in each column. In the special case of n = m, solutions either have exactly one vertex in each column or have exactly one vertex in each row.

			Array begins:
============================================
n\m | 0  1 2 3   4   5    6     7      8 ...
----+---------------------------------------
  0 | 1  1 1 1   1   1    1     1      1 ...
  1 | 1  0 1 3   6  10   15    21     28 ...
  2 | 1  1 4 3   4   5    6     7      8 ...
  3 | 1  3 3 6   4   5    6     7      8 ...
  4 | 1  6 4 4  80  65   96   133    176 ...
  5 | 1 10 5 5  65 410  306   427    568 ...
  6 | 1 15 6 6  96 306 5112  4207   6448 ...
  7 | 1 21 7 7 133 427 4207 48818  38424 ...
  8 | 1 28 8 8 176 568 6448 38424 695424 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A303211.
Column 0 is A000012.
Column 1 is A000217(n-1), n > 0.
Cf. A384116, A384118, A384119.

B(n,k) = {if(k<=n, if(k==1, binomial(n,2), sum(i=0, k, (-1)^i * binomial(k,i) * binomial(n,i) * i! * (n-i)^(k-i))))}
T(n,m) = {if(n==0&&m==0, 1, B(n,m) + B(m,n))}

T(n,m) = Sum_{i=0..m} (-1)^i * binomial(m,i) * binomial(n,i) * i! * (n-i)^(m-i) for 1 < m < n.
T(n,m) = T(m,n).
T(n,2) = T(n,3) = n for n >= 4.

A384118 Array read by antidiagonals: T(n,m) is the number of minimal total dominating sets in the n X m rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 6, 5, 5, 6, 1, 1, 10, 12, 51, 12, 10, 1, 1, 15, 37, 97, 97, 37, 15, 1, 1, 21, 98, 218, 368, 218, 98, 21, 1, 1, 28, 219, 519, 2229, 2229, 519, 219, 28, 1, 1, 36, 430, 1417, 6232, 7310, 6232, 1417, 430, 36, 1

Offset: 0

Andrew Howroyd, May 19 2025

			Array begins:
=====================================================
n\m | 0  1   2    3     4      5       6        7 ...
----+------------------------------------------------
  0 | 1  1   1    1     1      1       1        1 ...
  1 | 1  0   1    3     6     10      15       21 ...
  2 | 1  1   4    5    12     37      98      219 ...
  3 | 1  3   5   51    97    218     519     1417 ...
  4 | 1  6  12   97   368   2229    6232    16013 ...
  5 | 1 10  37  218  2229   7310   44491   172387 ...
  6 | 1 15  98  519  6232  44491  301572  1345693 ...
  7 | 1 21 219 1417 16013 172387 1345693 10893008 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A347921.

Cf. A290632, A384116, A384117.

B(n,m)={ my(M=matrix(n+1,m+1)); for(n=1, n, M[n+1,1]=1; for(m=1, m, M[n+1,m+1] = if(n>2, binomial(n,2)*M[n-1,m]) + sum(i=2, m, binomial(m-1,i-1)*(n*M[n, m-i+1] + if(i>=3&&i<=n, binomial(n,i-1)*i!*M[n-i+2,m-i+1] ) )))); M}
A(n,m)={ my(M=B(m,n) + B(n,m)~); M[1,1]=1; for(i=1, m, for(j=1, n, if((i+j)%3==0 && j<=2*i && i<=2*j, my(t=(i+j)/3); M[i+1,j+1] += binomial(i,j-t)*binomial(j,i-t)*(2*(j-t))!*(2*(i-t))!/2^t ))); M}
{ my(T=A(8,8)); for(i=1, #T, print(T[i, ])) }

T(n,m) = T(m,n).

cp's OEIS Frontend

A303208 Number of total dominating sets in the n X n rook graph.

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A384117 Array read by antidiagonals: T(n,m) is the number of minimum total dominating sets in the n X m rook graph K_n X K_m.

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A384118 Array read by antidiagonals: T(n,m) is the number of minimal total dominating sets in the n X m rook graph K_n X K_m.

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