A334772 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.
2, 12, 66, 36, 576, 1168, 80, 2610, 17376, 16220, 150, 8520, 129800, 448800, 202416, 252, 22680, 659560, 5748750, 10861056, 2395540, 392, 52416, 2596608, 46412200, 241987500, 253940736, 27517568, 576, 109116, 8505728, 273322980, 3121135440, 9885006250, 5807161344, 310123764
Offset: 2
Examples
Array begins: ========================================================== n\k | 2 3 4 5 ----|---------------------------------------------------- 2 | 2 12 36 80 ... 3 | 66 576 2610 8520 ... 4 | 1168 17376 129800 659560 ... 5 | 16220 448800 5748750 46412200 ... 6 | 202416 10861056 241987500 3121135440 ... 7 | 2395540 253940736 9885006250 203933233280 ... 8 | 27517568 5807161344 395426250000 13051880894720 ... ... The T(2,3) = 12 permutations of 111222 with 2 local maxima are 112122, 112212 and their rotations. The T(3,2) = 66 permutations of 112233 with 2 local maxima are 112323, 113223, 113232, 121233, 121332, 122133, 122313, 123213, 123123, 123132, 131322 and their rotations.
Links
- Andrew Howroyd, Table of n, a(n) for n = 2..1276
Crossrefs
Programs
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PARI
T(n,k)={n*k*( (k^2 + 4*k + 1)^2*binomial(k+3,3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2)}
Formula
T(n,k) = n*k*( P(k,4)^(n-2) * P(k-2,2)^2 + 4*(Sum_{j=0..n-3} P(k-1,3) * P(k-2,2) * P(k,2)^j * P(k, 4)^(n-j-3)) + 4*(Sum_{j=0..n-4} (j + 1) * P(k-1,3)^2 * P(k,2)^j * P(k,4)^(n-j-4)) )/2 where P(n,k) = binomial(n+k-1, k-1).
T(n,k) = n*k*( (k^2 + 4*k + 1)^2*binomial(k+3, 3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2).
Comments