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;
...
A276974
Number T(n,k) of permutations of [n] where the minimal 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, 4, 1, 1, 0, 19, 3, 1, 1, 0, 103, 12, 3, 1, 1, 0, 651, 54, 10, 3, 1, 1, 0, 4702, 281, 42, 10, 3, 1, 1, 0, 38413, 1652, 203, 37, 10, 3, 1, 1, 0, 350559, 11017, 1086, 166, 37, 10, 3, 1, 1, 0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1, 0, 39196758, 669948, 44265, 4900, 726, 151, 37, 10, 3, 1, 1
Offset: 0
T(3,1) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).
T(3,2) = 1: (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 1, 1;
0, 19, 3, 1, 1;
0, 103, 12, 3, 1, 1;
0, 651, 54, 10, 3, 1, 1;
0, 4702, 281, 42, 10, 3, 1, 1;
0, 38413, 1652, 203, 37, 10, 3, 1, 1;
0, 350559, 11017, 1086, 166, 37, 10, 3, 1, 1;
0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1;
...
A277032
Number of permutations of [n] such that the minimal cyclic distance between elements of the same cycle equals one, a(1)=1 by convention.
Original entry on oeis.org
1, 1, 5, 20, 109, 668, 4801, 38894, 353811, 3561512, 39374609, 474132730, 6179650125, 86676293916, 1301952953989, 20852719565694, 354771488612075, 6389625786835184, 121456993304945749, 2429966790591643402, 51042656559451380013, 1123165278137918510772
Offset: 1
a(2) = 1: (1,2).
a(3) = 5: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3), (1,3)(2).
-
b:= proc(n, i, l) option remember; `if`(n=0, mul(j!, j=l),
(m-> add(`if`(i=j or n*j=1, 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-1, 1, [0])):
seq(a(n), n=1..15);
-
b[n_, i_, l_] := b[n, i, l] = If[n == 0, Product[j!, {j, l}], With[{m = Length[l]}, Sum[If[i == j || n*j == 1, 0, b[n-1, j, If[j>m, Append[l, 0], ReplacePart[l, j -> l[[j]]+1]]]], {j, 1, m+1}]]];
a[n_] := If[n == 1, 1, n! - b[n-1, 1, {0}]];
Array[a, 15] (* Jean-François Alcover, Mar 13 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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