A094251 Number of one-element transitions between all set partitions of n labeled elements.

0, 0, 2, 18, 104, 580, 3282, 19236, 117672, 753048, 5041880, 35283402, 257718540, 1961679824, 15534932350, 127788932430, 1090212468512, 9632275777296, 88013486026710, 830637659785996, 8087069127986020, 81132805319035260, 837852685505824120, 8897619270153977254

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..70

Example

a(3) = 18 because there are 18 one-element transitions among the set partitions of n=3 elements ([x,z,y,...] means element 1 belongs to set x, element 2 belongs to set z, element 3 belongs to set y):
[1, 1, 1] -> [1, 1, 2]; [1, 1, 1] -> [1, 2, 1]; [1, 1, 1] -> [1, 2, 2];
[1, 1, 2] -> [1, 1, 1]; [1, 1, 2] -> [1, 2, 1]; [1, 1, 2] -> [1, 2, 2];
[1, 1, 2] -> [1, 2, 3]; [1, 2, 1] -> [1, 1, 1]; [1, 2, 1] -> [1, 1, 2];
[1, 2, 1] -> [1, 2, 2]; [1, 2, 1] -> [1, 2, 3]; [1, 2, 2] -> [1, 1, 1];
[1, 2, 2] -> [1, 1, 2]; [1, 2, 2] -> [1, 2, 1]; [1, 2, 2] -> [1, 2, 3];
[1, 2, 3] -> [1, 1, 2]; [1, 2, 3] -> [1, 2, 1]; [1, 2, 3] -> [1, 2, 2];

Maple

a:= proc(m) local b, r;
      b:= proc(n, i, p, l) local g, h, k;
            if i=0 then if n=0 then g:= l[1]; h:= l[2]; k:= l[3]+g+h;
                r:= r+p*(g*(g-1)/2+g*(k-g)+h*(1+2*(k-1))+(m-g-2*h)*k) fi
          else b(n, i-1, p, `if`(i<3, [0, l[]], l));
               seq(b(n-i*j, i-1, p*n!/(i!)^j/(n-i*j)!/j!,
                   `if`(i<3, [j, l[]], [l[]+j])), j=1..n/i)
            fi
          end;
      r:=0; b(m, max(m, 2), 1, [0]); r
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 13 2012

Mathematica

a[m_] := Module[{b, r}, b[n_, i_, p_, l_List] := Module[{g, h, k}, If[i == 0, If[n == 0, g = l[[1]]; h = l[[2]]; k = l[[3]] + g + h; r = r + p(g(g - 1)/2 + g(k - g) + h(1 + 2(k - 1)) + (m - g - 2h)k)], b[n, i - 1, p, If[i < 3, Prepend[l, 0], l]]; Table[b[n - i j, i - 1, p n!/(i!)^j/(n - i j)!/j!, If[i < 3, Prepend[l, j], l + j]], {j, 1, n/i}]]]; r = 0; b[m, Max[m, 2], 1, {0}]; r];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

Crossrefs

Cf. A093694, A093695.

Sequence in context: A267691 A219758 A005969 * A345969 A101570 A006043

Adjacent sequences: A094248 A094249 A094250 * A094252 A094253 A094254

Keyword

nonn

Author

Thomas Wieder, Apr 25 2004

Extensions

a(6)-a(23) from Alois P. Heinz, Apr 12 2012

Status

approved