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

0,5

Comments

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

Links

Alois P. Heinz, Rows n = 0..100, flattened

R. Stanley, Parking Functions, 2011

Formula

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

Example

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;

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

Crossrefs

Row sums give A265016.

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

Cf. A000217, A000272, A003056, A265208.

Sequence in context: A138057 A053727 A265208 * A325191 A209703 A279779

Adjacent sequences: A265017 A265018 A265019 * A265021 A265022 A265023

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 30 2015

Status

approved