A265208 Total number T(n,k) of lambda-parking functions induced by all partitions 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, 10, 0, 6, 14, 16, 0, 7, 21, 25, 0, 8, 27, 43, 0, 9, 36, 74, 0, 10, 44, 107, 125, 0, 11, 55, 146, 189, 0, 12, 65, 207, 307, 0, 13, 78, 267, 471, 0, 14, 90, 342, 786, 0, 15, 105, 436, 1058, 1296, 0, 16, 119, 538, 1490, 1921

Offset: 0

Author

Alois P. Heinz, Dec 04 2015

Keywords

• nonn
• tabf

Comments

Differs from A265020 first at T(5,2). See example.

Examples

			T(5,2) = 10: There are two partitions of 5 into 2 distinct parts: [2,3], [1,4]. Together they have 10 lambda-parking functions: [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1]. Here [1,1], [1,2], [1,3], [2,1], [3,1] are induced by both partitions. But they are counted only once.
T(6,1) = 6: [1], [2], [3], [4], [5], [6].
T(6,2) = 14: [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1].
T(6,3) = 16: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  10;
06 :  0,  6,  14,  16;
07 :  0,  7,  21,  25;
08 :  0,  8,  27,  43;
09 :  0,  9,  36,  74;
10 :  0, 10,  44, 107,  125;
11 :  0, 11,  55, 146,  189;
12 :  0, 12,  65, 207,  307;
13 :  0, 13,  78, 267,  471;
14 :  0, 14,  90, 342,  786;
15 :  0, 15, 105, 436, 1058, 1296;
16 :  0, 16, 119, 538, 1490, 1921;
		

Links

• Alois P. Heinz, Rows n = 0..400, flattened
• R. Stanley, Parking Functions, 2011

Crossrefs

Columns k=0-2 give: A000007, A000027, A176222(n+1).

Row sums give A265202.

Cf. A000217, A000272, A003056, A206735 (the same for general partitions), A265020, A265145.

Programs

Maple

b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
      `if`(n (p-> seq(coeff(p, x, i), i=0..degree(p)))(
        `if`(n=0, 1, b(0$2, n, 1$2))):
seq(T(n), n=0..25);

Mathematica

b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!*x^p] + If[nJean-François Alcover, Feb 02 2017, translated from Maple *)

Formula

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