A265016 - OEIS

%I #22 Aug 20 2021 04:26:59

%S 1,1,2,6,9,20,43,74,130,241,493,774,1413,2286,3987,7287,11650,19235,

%T 31581,50852,80867,141615,214538,349179,541603,859759,1303221,2054700,

%U 3277493,4960397,7652897,11662457,17703655,26603187,40043433,59384901,92234897,134538472

%N Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.

%H Alois P. Heinz, <a href="/A265016/b265016.txt">Table of n, a(n) for n = 0..100</a>

%H Richard P. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011.

%e The number of lambda-parking functions induced by the partitions of 4 into distinct parts:

%e 5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],

%e 4 by [4]: [1], [2], [3], [4].

%e a(4) = 5 + 4 = 9.

%p p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)

%p          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):

%p g:= (n, i, l)->  `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l),

%p     g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):

%p a:= n-> g(n$2, []):

%p seq(a(n), n=0..35);

%t p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,

%t      If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];

%t g[n_, i_, l_] := If[i (i + 1)/2 < n, 0, If[n == 0, p[l],

%t      g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];

%t a[n_] := If[n == 0, 1, g[n, n, {}]];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Aug 20 2021, after _Alois P. Heinz_ *)

%Y Row sums of A265017, A265018, A265019, A265020.

%Y Cf. A000009, A265007, A265202.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 30 2015