A332949
Number of entries in the ninth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.
Original entry on oeis.org
1, 91, 2531, 56717, 1052130, 17011450, 248006774, 3363718597, 43354519587, 537399621668, 6456347423794, 75743936924077, 874027443321519, 9978667891988711, 113225455087566673, 1281748270131892718, 14527578406583077101, 165413377044356558731
Offset: 9
-
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
end:
a:= n-> b(n, 1, 9)[2]:
seq(a(n), n=9..26);
A332950
Number of entries in the tenth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.
Original entry on oeis.org
1, 111, 3697, 97605, 2126580, 40204179, 681004277, 10645001317, 156970929310, 2213900198635, 30121302914917, 398061723460524, 5142929025812977, 65335359570066118, 819943536213362166, 10204014403455526051, 126342880437736660311, 1561117416681285339037
Offset: 10
-
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
end:
a:= n-> b(n, 1, 10)[2]:
seq(a(n), n=10..27);
A350202
Number T(n,k) of nodes in the k-th connected component of all endofunctions on [n] when components are ordered by increasing size; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 7, 1, 61, 19, 1, 709, 277, 37, 1, 9911, 4841, 811, 61, 1, 167111, 91151, 19706, 1876, 91, 1, 3237921, 1976570, 486214, 60229, 3739, 127, 1, 71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1, 1780353439, 1257567127, 380291461, 62248939, 5971291, 340729, 11197, 217, 1
Offset: 1
Triangle T(n,k) begins:
1;
7, 1;
61, 19, 1;
709, 277, 37, 1;
9911, 4841, 811, 61, 1;
167111, 91151, 19706, 1876, 91, 1;
3237921, 1976570, 486214, 60229, 3739, 127, 1;
71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1;
...
-
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(g(i)^j*
b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
(n, i$j, n-i*j)), j=0..n/i)))
end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..10);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][g[i]^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
T[n_, k_] := b[n, 1, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)