A285793
Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 4, 2, 18, 13, 5, 96, 83, 43, 18, 600, 582, 342, 192, 84, 4320, 4554, 2874, 1824, 1068, 480, 35280, 39672, 26232, 17832, 11784, 7080, 3240, 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200, 3265920, 4044240, 2834640, 2078640, 1534320, 1110960, 765360, 473760, 221760
Offset: 1
T(3,2) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.
Triangle T(n,k) begins:
: 1;
: 4, 2;
: 18, 13, 5;
: 96, 83, 43, 18;
: 600, 582, 342, 192, 84;
: 4320, 4554, 2874, 1824, 1068, 480;
: 35280, 39672, 26232, 17832, 11784, 7080, 3240;
: 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;
Main diagonal and first lower diagonal give:
A038720(n-1) (for n>1),
A286175.
A285382
Sum of entries in the last cycles of all permutations of [n].
Original entry on oeis.org
1, 5, 25, 143, 942, 7074, 59832, 563688, 5858640, 66622320, 823055040, 10979133120, 157300375680, 2409321801600, 39290164300800, 679701862425600, 12433400027596800, 239791474805299200, 4863054420016128000, 103462238924835840000, 2304147629440419840000
Offset: 1
a(3) = 25 because the sum of the entries in the last cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+2+5+3 = 25.
-
a:= proc(n) option remember; `if`(n<3, n*(3*n-1)/2,
((2*n^2+3*n-1)*a(n-1)-(n+2)*(n-1)*n*a(n-2))/(n+1))
end:
seq(a(n), n=1..25);
-
Table[n! * (n-1 + 2*(n+1)*HarmonicNumber[n])/4, {n, 1, 25}] (* Vaclav Kotesovec, Apr 29 2017 *)
A286232
Sum T(n,k) of the entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 5, 1, 19, 10, 1, 75, 57, 17, 1, 323, 285, 145, 26, 1, 1512, 1421, 975, 317, 37, 1, 7630, 7395, 5999, 2865, 616, 50, 1, 41245, 40726, 36183, 22411, 7315, 1094, 65, 1, 237573, 237759, 221689, 163488, 72581, 16630, 1812, 82, 1, 1451359, 1468162, 1405001, 1160764, 649723, 206249, 34425, 2840, 101, 1
Offset: 1
T(3,2) = 10 because the sum of the entries in the second last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+4+1+2 = 10.
Triangle T(n,k) begins:
1;
5, 1;
19, 10, 1;
75, 57, 17, 1;
323, 285, 145, 26, 1;
1512, 1421, 975, 317, 37, 1;
7630, 7395, 5999, 2865, 616, 50, 1;
41245, 40726, 36183, 22411, 7315, 1094, 65, 1;
...
-
app[P_, n_] := Module[{P0}, Table[P0 = Append[P, {}]; AppendTo[P0[[i]], n]; If[Last[P0] == {}, Most[P0], P0], {i, 1, Length[P]+1}]];
setPartitions[n_] := setPartitions[n] = If[n == 1, {{{1}}}, Flatten[app[#, n]& /@ setPartitions[n-1], 1]];
T[n_, k_] := Select[setPartitions[n], Length[#] >= k&][[All, -k]] // Flatten // Total;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 21 2021 *)
Showing 1-3 of 3 results.
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