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A334772
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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.
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7
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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
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OFFSET
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2,1
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COMMENTS
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T(n,k) is divisible by n and 2*T(n,k) is divisible by n*k.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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PROG
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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)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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