A265017 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with smallest part k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 5, 0, 0, 4, 0, 7, 8, 0, 0, 5, 0, 25, 12, 0, 0, 0, 6, 0, 36, 16, 15, 0, 0, 0, 7, 0, 81, 20, 21, 0, 0, 0, 0, 8, 0, 107, 74, 27, 24, 0, 0, 0, 0, 9, 0, 316, 102, 33, 32, 0, 0, 0, 0, 0, 10, 0, 427, 222, 39, 40, 35, 0, 0, 0, 0, 0, 11

Offset: 0

Author

Alois P. Heinz, Nov 30 2015

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
00 :  1;
01 :  0,   1;
02 :  0,   0,   2;
03 :  0,   3,   0,   3;
04 :  0,   5,   0,   0,  4;
05 :  0,   7,   8,   0,  0,  5;
06 :  0,  25,  12,   0,  0,  0, 6;
07 :  0,  36,  16,  15,  0,  0, 0, 7;
08 :  0,  81,  20,  21,  0,  0, 0, 0, 8;
09 :  0, 107,  74,  27, 24,  0, 0, 0, 0, 9;
10 :  0, 316, 102,  33, 32,  0, 0, 0, 0, 0, 10;
11 :  0, 427, 222,  39, 40, 35, 0, 0, 0, 0,  0, 11;
12 :  0, 869, 286, 153, 48, 45, 0, 0, 0, 0,  0,  0, 12;
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• Richard P. Stanley, Parking Functions, 2011.

Crossrefs

Row sums give A265016.

Column k=0 gives A000007.

Main diagonal gives A028310, first lower diagonal is A000004.

T(2n+1,n) gives A005563.

T(2n+2,n) gives A028347(n+2).

T(2n+3,n) gives A028560.

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[]])))):
T:= n-> (f-> seq(coeff(f, x, i), i=0..n))(g(n$2, [])):
seq(T(n), n=0..16);

Mathematica

p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,
     If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];
g[n_, i_, l_] := If[i(i+1)/2 < n, 0, If[n == 0, p[l]*x^
     If[l == {}, 0, l[[1]]], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)