A334773 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n with exactly 2 local maxima.
3, 12, 57, 30, 360, 705, 60, 1400, 7968, 7617, 105, 4170, 51750, 163584, 78357, 168, 10437, 241080, 1830000, 3293184, 791589, 252, 23072, 894201, 13562040, 64168750, 65968128, 7944321, 360, 46440, 2804480, 75278553, 759940800, 2246625000, 1319854080, 79541625
Offset: 2
Examples
Array begins: ====================================================== n\k | 2 3 4 5 ----|------------------------------------------------- 2 | 3 12 30 60 ... 3 | 57 360 1400 4170 ... 4 | 705 7968 51750 241080 ... 5 | 7617 163584 1830000 13562040 ... 6 | 78357 3293184 64168750 759940800 ... 7 | 791589 65968128 2246625000 42560067360 ... 8 | 7944321 1319854080 78636093750 2383387566720 ... ... The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.
Links
- Andrew Howroyd, Table of n, a(n) for n = 2..1276
Crossrefs
Programs
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PARI
T(n,k) = {3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2}
Formula
T(n,k) = Sum_{j=0..n-2} P(k-1,3) * P(k-2,2) * P(k,2)^(n-2-j) * P(k,4)^j + 2 * (n-j-2) * P(k-1,3)^2 * P(k,2)^(n-3-j) * P(k,4)^j where P(n,k) = binomial(n+k-1,k-1).
T(n,k) = 3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2.