A285439
Sum T(n,k) of the entries in the k-th 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, 21, 12, 3, 132, 76, 28, 4, 960, 545, 235, 55, 5, 7920, 4422, 2064, 612, 96, 6, 73080, 40194, 19607, 6692, 1386, 154, 7, 745920, 405072, 202792, 75944, 18736, 2816, 232, 8, 8346240, 4484808, 2280834, 911637, 254061, 46422, 5256, 333, 9
Offset: 1
T(3,1) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
Triangle T(n,k) begins:
1;
4, 2;
21, 12, 3;
132, 76, 28, 4;
960, 545, 235, 55, 5;
7920, 4422, 2064, 612, 96, 6;
73080, 40194, 19607, 6692, 1386, 154, 7;
745920, 405072, 202792, 75944, 18736, 2816, 232, 8;
...
Main diagonal and first lower diagonal give:
A000027,
A006000 (for n>0).
-
T:= proc(h) option remember; local b; b:=
proc(n, l) option remember; `if`(n=0, [mul((i-1)!, i=l), 0],
(p-> p+[0, (h-n+1)*p[1]*x^(nops(l)+1)])(b(n-1, [l[], 1]))+
add((p-> p+[0, (h-n+1)*p[1]*x^j])(
b(n-1, subsop(j=l[j]+1, l))), j=1..nops(l)))
end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
end:
seq(T(n), n=1..10);
-
T[h_] := T[h] = Module[{b}, b[n_, l_] := b[n, l] = If[n == 0, {Product[(i - 1)!, {i, l}], 0}, # + {0, (h - n + 1)*#[[1]]*x^(Length[l] + 1)}&[b[n - 1, Append[l, 1]]] + Sum[# + {0, (h-n+1)*#[[1]]*x^j}&[b[n - 1, ReplacePart[ l, j -> l[[j]] + 1]]], {j, 1, Length[l]}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, {}][[2]]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)
A284816
Sum of entries in the first cycles of all permutations of [n].
Original entry on oeis.org
1, 4, 21, 132, 960, 7920, 73080, 745920, 8346240, 101606400, 1337212800, 18920563200, 286442956800, 4620449433600, 79114299264000, 1433211107328000, 27387931963392000, 550604138692608000, 11617107089043456000, 256671161862635520000, 5926549291918295040000
Offset: 1
a(3) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
-
a:= n-> n!*(n*(n+1)-(n-1)*(n+2)/2)/2:
seq(a(n), n=1..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
(n^2+n+2)*n*a(n-1)/(n^2-n+2))
end:
seq(a(n), n=1..25);
A285424
Sum of the entries in the last blocks of all set partitions of [n].
Original entry on oeis.org
1, 5, 19, 75, 323, 1512, 7630, 41245, 237573, 1451359, 9365361, 63604596, 453206838, 3378581609, 26285755211, 212953670251, 1792896572319, 15658150745252, 141619251656826, 1324477898999161, 12791059496663293, 127395689514237279, 1307010496324272157
Offset: 1
a(3) = 19 because the sum of the entries in the last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 6+3+2+5+3 = 19.
-
a:= proc(h) option remember; local b; b:=
proc(n, m, s) option remember; `if`(n=0, s,
add(b(n-1, max(m, j), `if`(j
-
a[h_] := a[h] = Module[{b}, b[n_, m_, s_] := b[n, m, s] = If[n == 0, s, Sum[b[n-1, Max[m, j], If[j < m, s, h - n + 1 + If[j == m, s, 0]]], {j, 1, m + 1}]]; b[h, 0, 0]];
Array[a, 25] (* Jean-François Alcover, May 22 2018, translated from Maple *)
A286231
Sum T(n,k) of the entries in the k-th last cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 5, 1, 25, 10, 1, 143, 79, 17, 1, 942, 634, 197, 26, 1, 7074, 5462, 2129, 417, 37, 1, 59832, 51214, 23381, 5856, 786, 50, 1, 563688, 523386, 269033, 80053, 13934, 1360, 65, 1, 5858640, 5813892, 3281206, 1111498, 232349, 29728, 2204, 82, 1
Offset: 1
T(3,2) = 10 because the sum of the entries in the second last cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 0+0+3+4+1+2 = 10.
Triangle T(n,k) begins:
1;
5, 1;
25, 10, 1;
143, 79, 17, 1;
942, 634, 197, 26, 1;
7074, 5462, 2129, 417, 37, 1;
59832, 51214, 23381, 5856, 786, 50, 1;
563688, 523386, 269033, 80053, 13934, 1360, 65, 1;
...
Showing 1-4 of 4 results.
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