A263757
Triangle read by rows: T(n,k) (n>=1, 0<=k
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 4, 7, 12, 1, 7, 17, 35, 60, 1, 12, 44, 93, 210, 360, 1, 20, 103, 275, 651, 1470, 2520, 1, 33, 234, 877, 2047, 5208, 11760, 20160, 1, 54, 533, 2544, 7173, 18423, 46872, 105840, 181440, 1, 88, 1196, 7135, 27085, 67545, 184230, 468720, 1058400, 1814400
Offset: 1
Triangle begins:
1;
1, 1;
1, 2, 3;
1, 4, 7, 12;
1, 7, 17, 35, 60;
1, 12, 44, 93, 210, 360;
...
A276975
Number of permutations of [n] such that the minimal distance between elements of the same cycle equals one, a(1)=1 by convention.
Original entry on oeis.org
1, 1, 4, 19, 103, 651, 4702, 38413, 350559, 3539511, 39196758, 472612883, 6165080443, 86526834271, 1300282224846, 20832761552453, 354515666646827, 6386139146435035, 121406489336263622, 2429193186525638435, 51030147426536745655, 1122952442325988152627
Offset: 1
a(2) = 1: (1,2).
a(3) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).
-
b:= proc(n, i, l) option remember; `if`(n=0, mul(j!, j=l),
(m-> add(`if`(i=j, 0, b(n-1, j, `if`(j>m, [l[], 0],
subsop(j=l[j]+1, l)))), j=1..m+1))(nops(l)))
end:
a:= n-> `if`(n=1, 1, n!-b(n, 0, [])):
seq(a(n), n=1..15);
-
b[n_, i_, l_] := b[n, i, l] = If[n == 0, Product[j!, {j, l}], Function[m, Sum[If[i == j, 0, b[n - 1, j, If[j > m, Append[l, 0], ReplacePart[l, j -> l[[j]] + 1]]]], {j, 1, m + 1}]][Length[l]]];
a[n_] := If[n == 1, 1, n! - b[n, 0, {}]];
Array[a, 15] (* Jean-François Alcover, Oct 28 2020, after Maple code *)
A277031
Number T(n,k) of permutations of [n] where the minimal cyclic distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 20, 3, 0, 1, 0, 109, 10, 0, 0, 1, 0, 668, 44, 7, 0, 0, 1, 0, 4801, 210, 28, 0, 0, 0, 1, 0, 38894, 1320, 90, 15, 0, 0, 0, 1, 0, 353811, 8439, 554, 75, 0, 0, 0, 0, 1, 0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1, 0, 39374609, 517418, 23298, 1276, 198, 0, 0, 0, 0, 0, 1
Offset: 0
T(3,1) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 5, 0, 1;
0, 20, 3, 0, 1;
0, 109, 10, 0, 0, 1;
0, 668, 44, 7, 0, 0, 1;
0, 4801, 210, 28, 0, 0, 0, 1;
0, 38894, 1320, 90, 15, 0, 0, 0, 1;
0, 353811, 8439, 554, 75, 0, 0, 0, 0, 1;
0, 3561512, 63404, 3542, 310, 31, 0, 0, 0, 0, 1;
...
Showing 1-3 of 3 results.
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