A235942
Number of positions (cyclic permutations) of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals counted as distinct.
Original entry on oeis.org
0, 0, 0, 0, 50, 144, 1078, 7936, 66096, 611200, 6248682, 69926976, 850848414, 11187719984, 158122436400, 2390945284096, 38518483536706, 658706393035152, 11918123304961222, 227474585229393600, 4567806759318652080
Offset: 1
- Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences
A165960
Number of permutations of length n without modular 3-sequences.
Original entry on oeis.org
1, 1, 2, 3, 20, 100, 612, 4389, 35688, 325395, 3288490, 36489992, 441093864, 5770007009, 81213878830, 1223895060315, 19662509071056, 335472890422812, 6057979285535388, 115434096553014565, 2314691409652237700, 48723117262650147387, 1074208020519710570054
Offset: 0
For n=3 the a(3) = 3 solutions are (0,2,1), (1,0,2) and (2,1,0).
A372102
Number of permutations of [n] whose non-fixed points are not neighbors.
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 19, 45, 107, 278, 728, 2033, 5749, 17105, 51669, 162674, 520524, 1724329, 5807143, 20146861, 71048431, 257139686, 945626800, 3558489633, 13599579817, 53060155137, 210124405097, 847904374466, 3470756061140, 14453943647561, 61023690771451
Offset: 0
a(3) = 2: 123, 321.
a(4) = 4: 1234, 1432, 3214, 4231.
a(5) = 9: 12345, 12543, 14325, 15342, 32145, 32541, 42315, 52143, 52341.
a(6) = 19: 123456, 123654, 125436, 126453, 143256, 143652, 153426, 163254, 163452, 321456, 325416, 326451, 423156, 423651, 521436, 523416, 621453, 623154, 623451.
-
a:= proc(n) option remember; `if`(n<4, [2$3, 4][n+1],
3*a(n-1)+(n-2)*a(n-2)+(n-1)*(a(n-4)-a(n-3)))/2
end:
seq(a(n), n=0..30);
-
a[n_] := Sum[Binomial[n + 1 - k, k] * Subfactorial[k], {k, 0, (n + 1)/2}];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Apr 24 2024 *)
A180186
Triangle read by rows: T(n,k) is the number of permutations of [n] starting with 1, having no 3-sequences and having k successions (0 <= k <= floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 2, 3, 0, 9, 8, 3, 44, 45, 12, 1, 265, 264, 90, 8, 1854, 1855, 660, 90, 2, 14833, 14832, 5565, 880, 45, 133496, 133497, 51912, 9275, 660, 9, 1334961, 1334960, 533988, 103824, 9275, 264, 14684570, 14684571, 6007320, 1245972, 129780, 5565
Offset: 0
T(5,2)=3 because we have 12453, 12534, and 14523.
Triangle starts:
1;
1;
0, 1;
1, 0;
2, 3, 0;
9, 8, 3;
44, 45, 12, 1;
265, 264, 90, 8;
-
d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(n-k, k)*d[n-1-k] else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form
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