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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.

%I A265208 #29 Feb 02 2017 10:02:07
%S A265208 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,
%T A265208 0,10,44,107,125,0,11,55,146,189,0,12,65,207,307,0,13,78,267,471,0,14,
%U A265208 90,342,786,0,15,105,436,1058,1296,0,16,119,538,1490,1921
%N 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.
%C A265208 Differs from A265020 first at T(5,2). See example.
%H A265208 Alois P. Heinz, <a href="/A265208/b265208.txt">Rows n = 0..400, flattened</a>
%H A265208 R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011
%F A265208 T(A000217(n),n) = A000272(n+1).
%e A265208 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.
%e A265208 T(6,1) = 6: [1], [2], [3], [4], [5], [6].
%e A265208 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].
%e A265208 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].
%e A265208 Triangle T(n,k) begins:
%e A265208 00 :  1;
%e A265208 01 :  0,  1;
%e A265208 02 :  0,  2;
%e A265208 03 :  0,  3,   3;
%e A265208 04 :  0,  4,   5;
%e A265208 05 :  0,  5,  10;
%e A265208 06 :  0,  6,  14,  16;
%e A265208 07 :  0,  7,  21,  25;
%e A265208 08 :  0,  8,  27,  43;
%e A265208 09 :  0,  9,  36,  74;
%e A265208 10 :  0, 10,  44, 107,  125;
%e A265208 11 :  0, 11,  55, 146,  189;
%e A265208 12 :  0, 12,  65, 207,  307;
%e A265208 13 :  0, 13,  78, 267,  471;
%e A265208 14 :  0, 14,  90, 342,  786;
%e A265208 15 :  0, 15, 105, 436, 1058, 1296;
%e A265208 16 :  0, 16, 119, 538, 1490, 1921;
%p A265208 b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
%p A265208       `if`(n<t, 0, add(b(p+1, `if`(i=j, g+1, 1), n-max(j, t), j,
%p A265208        max(j, t)+1)/`if`(i=j, 1, g!), j=i..n))
%p A265208     end:
%p A265208 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p A265208         `if`(n=0, 1, b(0$2, n, 1$2))):
%p A265208 seq(T(n), n=0..25);
%t A265208 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 *)
%Y A265208 Columns k=0-2 give: A000007, A000027, A176222(n+1).
%Y A265208 Row sums give A265202.
%Y A265208 Cf. A000217, A000272, A003056, A206735 (the same for general partitions), A265020, A265145.
%K A265208 nonn,tabf
%O A265208 0,5
%A A265208 _Alois P. Heinz_, Dec 04 2015