A350202 - OEIS

%I #17 Mar 18 2022 04:32:56

%S 1,7,1,61,19,1,709,277,37,1,9911,4841,811,61,1,167111,91151,19706,

%T 1876,91,1,3237921,1976570,486214,60229,3739,127,1,71850913,47203241,

%U 13110749,1892997,152937,6721,169,1,1780353439,1257567127,380291461,62248939,5971291,340729,11197,217,1

%N 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.

%H Alois P. Heinz, <a href="/A350202/b350202.txt">Rows n = 1..141, flattened</a>

%e Triangle T(n,k) begins:

%e          1;

%e          7,        1;

%e         61,       19,        1;

%e        709,      277,       37,       1;

%e       9911,     4841,      811,      61,      1;

%e     167111,    91151,    19706,    1876,     91,    1;

%e    3237921,  1976570,   486214,   60229,   3739,  127,   1;

%e   71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1;

%e   ...

%p g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:

%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,

%p       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(g(i)^j*

%p         b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]

%p          (n, i$j, n-i*j)), j=0..n/i)))

%p     end:

%p T:= (n, k)-> b(n, 1, k)[2]:

%p seq(seq(T(n, k), k=1..n), n=1..10);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];

%t 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 T[n_, k_] := b[n, 1, k][[2]];

%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)

%Y Column k=1 gives A350157.

%Y Row sums give A007778.

%Y T(n+1,n) gives A003215 for n>=1.

%Y Cf. A001865, A319298, A322383.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, Dec 19 2021