A265130 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of k into distinct parts with largest part n and n<=k<=n*(n+1)/2.

1, 1, 5, 32, 272, 2957, 39531, 629806, 11673074, 247028567, 5881190801, 155651692748, 4534744862052, 144246963009697, 4975152075900887, 184958685188293274, 7373625038400716198, 313817002976857310507, 14201832585602869616349, 681022860320979979626232

Offset: 0

Author

Alois P. Heinz, Dec 02 2015

Keywords

• nonn

Links

• R. Stanley, Parking Functions, 2011

Crossrefs

Column sums of A265018, A265019.

Programs

Maple

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)-> `if`(i*(i+1)/2n, 0, g(n-i, i-1, [i, l[]])))):
a:= n-> `if`(n=0, 1, add(g(k-n, n-1, [n]), k=n..n*(n+1)/2)):
seq(a(n), n=0..10);

Mathematica

p[l_] := Function[n, n!*Det[Table[Function [t,
     If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]][Length[l]];
g[n_, i_, l_] := If[i(i+1)/2 < n, 0,
     If[n == 0, p[l], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
a[n_] := If[n == 0, 1, Sum[g[k - n, n - 1, {n}], {k, n, n(n+1)/2}]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Aug 22 2021, after Alois P. Heinz *)