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

0,5

Comments

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

Links

Alois P. Heinz, Rows n = 0..400, flattened

R. Stanley, Parking Functions, 2011

Formula

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

Example

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;

Maple

b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
      `if`(n<t, 0, add(b(p+1, `if`(i=j, g+1, 1), n-max(j, t), j,
       max(j, t)+1)/`if`(i=j, 1, g!), j=i..n))
    end:
T:= 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[n<t, 0, Sum[b[p+1, If[i==j, g+1, 1], n-Max[j, t], j, Max[j, t]+1] / If[i==j, 1, g!], {j, i, n}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][If[n==0, 1, b[0, 0, n, 1, 1]]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

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.

Sequence in context: A328911 A138057 A053727 * A265020 A325191 A209703

Adjacent sequences: A265205 A265206 A265207 * A265209 A265210 A265211

Keyword

nonn,tabf

Author

Alois P. Heinz, Dec 04 2015

Status

approved