A384124 Array read by antidiagonals: T(n,m) is the number of irredundant sets in the n X m rook complement graph.
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 9, 8, 1, 1, 16, 24, 24, 16, 1, 1, 32, 77, 94, 77, 32, 1, 1, 64, 178, 284, 284, 178, 64, 1, 1, 128, 373, 624, 777, 624, 373, 128, 1, 1, 256, 724, 1234, 1620, 1620, 1234, 724, 256, 1, 1, 512, 1331, 2258, 3049, 3286, 3049, 2258, 1331, 512, 1
Offset: 0
Examples
Array begins: =============================================== n\m | 0 1 2 3 4 5 6 7 ... ----+------------------------------------------ 0 | 1 1 1 1 1 1 1 1 ... 1 | 1 2 4 8 16 32 64 128 ... 2 | 1 4 9 24 77 178 373 724 ... 3 | 1 8 24 94 284 624 1234 2258 ... 4 | 1 16 77 284 777 1620 3049 5332 ... 5 | 1 32 178 624 1620 3286 6022 10268 ... 6 | 1 64 373 1234 3049 6022 10771 17962 ... 7 | 1 128 724 2258 5332 10268 17962 29366 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
- Eric Weisstein's World of Mathematics, Irredundant Set.
- Eric Weisstein's World of Mathematics, Rook Complement Graph.
Programs
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PARI
T(n,m) = {n*(2^m-1) + m*(2^n-1) - n*m + if(n>2&&m>2,6,if(n+m>4, 2))*binomial(n,2)*binomial(m,2) + 6*binomial(n,3)*binomial(m,3) + if(m>3,6*binomial(n,2)*binomial(m,3)) + if(n>3,6*binomial(n,3)*binomial(m,2)) + 6*binomial(n,4)*binomial(m,2) + 6*binomial(n,2)*binomial(m,4) + 1}
Formula
T(n,m) = n*(2^m-1) + m*(2^n-1) - n*m + binomial(n,2)*binomial(m,2) + 6*binomial(n,3)*binomial(m,3) + 6*binomial(n,2)*binomial(m,3) + 6*binomial(n,3)*binomial(m,2) + 6*binomial(n,4)*binomial(m,2) + 6*binomial(n,2)*binomial(m,4) + 1 for n >= 4, m >= 4.