A265020 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 2, 0, 3, 3, 0, 4, 5, 0, 5, 15, 0, 6, 21, 16, 0, 7, 42, 25, 0, 8, 54, 68, 0, 9, 90, 142, 0, 10, 110, 248, 125, 0, 11, 165, 409, 189, 0, 12, 195, 710, 496, 0, 13, 273, 1033, 967, 0, 14, 315, 1562, 2096, 0, 15, 420, 2291, 3265, 1296, 0, 16, 476, 3180

Offset: 0

Author

Alois P. Heinz, Nov 30 2015

Keywords

• nonn
• tabf

Comments

Differs from A265208 first at T(5,2). See example.

Examples

			T(5,2) = 15 because there are two partitions of 5 into 2 distinct parts: [2,3] and [1,4]. And [2,3] has 8 lambda-parking functions: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2] and [1,4] has 7: [1,1], [1,2], [1,3], [1,4], [2,1], [3,1], [4,1]. So [1,1], [1,2], [1,3], [2,1], [3,1] are counted twice.
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  15;
06 :  0,  6,  21,   16;
07 :  0,  7,  42,   25;
08 :  0,  8,  54,   68;
09 :  0,  9,  90,  142;
10 :  0, 10, 110,  248,  125;
11 :  0, 11, 165,  409,  189;
12 :  0, 12, 195,  710,  496;
13 :  0, 13, 273, 1033,  967;
14 :  0, 14, 315, 1562, 2096;
15 :  0, 15, 420, 2291, 3265, 1296;
16 :  0, 16, 476, 3180, 6057, 1921;
		

Links

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

Crossrefs

Row sums give A265016.

Columns k=0-1 give: A000007, A000027.

Cf. A000217, A000272, A003056, A265208.

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..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);

Mathematica

p[l_] := With[{n = Length[l]}, n!*Det[Table[With[{t = j - i + 1}, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]];
g[n_, i_, l_] := If[i*(i + 1)/2 < n, 0, If[n == 0, p[l]*x^Length[l], g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Join[{i}, l]]]]];
T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jul 29 2024, after Alois P. Heinz *)

Formula

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