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

0,6

Links

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

Richard P. Stanley, Parking Functions, 2011.

Example

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;

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^
                `if`(l=[], 0, l[1]), 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..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 *)

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.

Sequence in context: A161123 A035442 A213177 * A349136 A035376 A259708

Adjacent sequences: A265014 A265015 A265016 * A265018 A265019 A265020

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 30 2015

Status

approved