A265019 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

1, 1, 2, 3, 3, 5, 8, 16, 4, 7, 12, 40, 34, 50, 125, 5, 9, 16, 55, 73, 132, 281, 351, 307, 432, 1296, 6, 11, 20, 70, 96, 212, 469, 642, 1020, 1361, 3294, 3305, 3910, 3506, 4802, 16807, 7, 13, 24, 85, 119, 267, 644, 959, 1567, 2686, 5570, 7109, 11890, 13234

Offset: 0

Author

Alois P. Heinz, Nov 30 2015

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
00 :  1;
01 :     1;
02 :        2;
03 :        3,  3;
04 :            5,   4;
05 :            8,   7,   5;
06 :           16,  12,   9,   6;
07 :                40,  16,  11,   7;
08 :                34,  55,  20,  13,   8;
09 :                50,  73,  70,  24,  15,   9;
10 :               125, 132,  96,  85,  28,  17, 10;
11 :                    281, 212, 119, 100,  32, 19, 11;
12 :                    351, 469, 267, 142, 115, 36, 21, 12;
		

Links

• Alois P. Heinz, Columns k = 0..22, flattened
• R. Stanley, Parking Functions, 2011.

Crossrefs

Row sums give A265016.

Column sums give A265130.

Cf. A000217, A000272, A265018 (the same read by rows).

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[]])))):
b:= proc(n) option remember; g(n$2, []) end:
T:= k-> seq(coeff(b(n), x, k), n=k..k*(k+1)/2):
seq(T(k), k=0..8);

Mathematica

p[l_] := With[{n = Length[l]}, n!*Det[Table[t = j-i+1; If[t<0, 0, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[i*(i+1)/2n, 0, g[n-i, i-1, Join[{i}, l]]]]]; b[n_] := b[n] = g[n, n, {}]; T[0] = {1}; T[k_] := Table[Coefficient[b[n], x, k], {n, k, k*(k+1)/2}]; Table[T[k], {k, 0, 8}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)

Formula

T(A000217(n),n) = A000272(n+1).