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;
...
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;
...
A239144
Number T(n,k) of self-inverse permutations p on [n] such that all transposition distances (if any) are larger than k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 10, 5, 2, 1, 1, 26, 13, 5, 2, 1, 1, 76, 37, 15, 5, 2, 1, 1, 232, 112, 47, 15, 5, 2, 1, 1, 764, 363, 155, 52, 15, 5, 2, 1, 1, 2620, 1235, 532, 188, 52, 15, 5, 2, 1, 1, 9496, 4427, 1910, 704, 203, 52, 15, 5, 2, 1, 1
Offset: 0
T(4,0) = 10: 1234, 1243, 1324, 1432, 2134, 2143, 3214, 3412, 4231, 4321.
T(4,1) = 5: 1234, 1432, 3214, 3412, 4231.
T(4,2) = 2: 1234, 4231.
T(4,3) = 1: 1234.
Triangle T(n,k) begins:
00: 1;
01: 1, 1;
02: 2, 1, 1;
03: 4, 2, 1, 1;
04: 10, 5, 2, 1, 1;
05: 26, 13, 5, 2, 1, 1;
06: 76, 37, 15, 5, 2, 1, 1;
07: 232, 112, 47, 15, 5, 2, 1, 1;
08: 764, 363, 155, 52, 15, 5, 2, 1, 1;
09: 2620, 1235, 532, 188, 52, 15, 5, 2, 1, 1;
10: 9496, 4427, 1910, 704, 203, 52, 15, 5, 2, 1, 1;
-
b:= proc(n, k, s) option remember; `if`(n=0, 1, `if`(n in s,
b(n-1, k, s minus {n}), b(n-1, k, s) +add(`if`(i in s, 0,
b(n-1, k, s union {i})), i=1..n-k-1)))
end:
T:= (n, k)-> b(n, k, {}):
seq(seq(T(n, k), k=0..n), n=0..14);
-
b[n_, k_, s_List] := b[n, k, s] = If[n == 0, 1, If[MemberQ[s, n], b[n-1, k, s ~Complement~ {n}], b[n-1, k, s] + Sum[If[MemberQ[s, i], 0, b[n-1, k, s ~Union~ {i}]], {i, 1, n-k-1}]]]; T[n_, k_] := b[n, k, {}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
Showing 1-3 of 3 results.
Comments