A290818 Array read by antidiagonals: T(m,n) = number of irredundant sets in the lattice (rook) graph K_m X K_n.
2, 3, 3, 4, 11, 4, 5, 24, 24, 5, 6, 47, 94, 47, 6, 7, 88, 272, 272, 88, 7, 8, 163, 774, 1185, 774, 163, 8, 9, 304, 2230, 4280, 4280, 2230, 304, 9, 10, 575, 6542, 15781, 20106, 15781, 6542, 575, 10, 11, 1104, 19452, 60604, 88512, 88512, 60604, 19452, 1104, 11
Offset: 1
Examples
Array begins: =============================================================== m\n| 1 2 3 4 5 6 7 8 ---+----------------------------------------------------------- 1 | 2 3 4 5 6 7 8 9 ... 2 | 3 11 24 47 88 163 304 575 ... 3 | 4 24 94 272 774 2230 6542 19452 ... 4 | 5 47 272 1185 4280 15781 60604 240073 ... 5 | 6 88 774 4280 20106 88512 400728 1879744 ... 6 | 7 163 2230 15781 88512 453271 2326534 12363513 ... 7 | 8 304 6542 60604 400728 2326534 13169346 76446456 ... 8 | 9 575 19452 240073 1879744 12363513 76446456 476777153 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
- Eric Weisstein's World of Mathematics, Irredundant Set
- Eric Weisstein's World of Mathematics, Rook Graph
Programs
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Mathematica
s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}]; c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}]; p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}]; b[m_, n_, x_]:=m^n*x^n + n^m*x^m - If[n<=m, n!*x^m*StirlingS2[m, n], m!*x^n*StirlingS2[n, m]]; T[m_, n_]:= b[m, n, 1] + p[m, n, 1]; Table[T[n, m -n + 1], {m, 10}, {n, m}]//Flatten (* Indranil Ghosh, Aug 12 2017, after PARI code *) -
PARI
\\ See A. Howroyd note in A290586 for explanation. s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) ); c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i); p(m,n,x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m, k) * binomial(n, r) * k! * s(r, k) * x^r * c(m-k, n-r, x) ))} b(m,n,x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2)); T(m,n) = b(m,n,1) + p(m,n,1); for(m=1,10,for(n=1,m,print1(T(n,m-n+1),", ")));