A265020 - OEIS

%I #22 Jul 29 2024 07:13:36

%S 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,

%T 142,0,10,110,248,125,0,11,165,409,189,0,12,195,710,496,0,13,273,1033,

%U 967,0,14,315,1562,2096,0,15,420,2291,3265,1296,0,16,476,3180

%N 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.

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

%H Alois P. Heinz, <a href="/A265020/b265020.txt">Rows n = 0..100, flattened</a>

%H R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011

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

%e 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.

%e Triangle T(n,k) begins:

%e 00 :  1;

%e 01 :  0,  1;

%e 02 :  0,  2;

%e 03 :  0,  3,   3;

%e 04 :  0,  4,   5;

%e 05 :  0,  5,  15;

%e 06 :  0,  6,  21,   16;

%e 07 :  0,  7,  42,   25;

%e 08 :  0,  8,  54,   68;

%e 09 :  0,  9,  90,  142;

%e 10 :  0, 10, 110,  248,  125;

%e 11 :  0, 11, 165,  409,  189;

%e 12 :  0, 12, 195,  710,  496;

%e 13 :  0, 13, 273, 1033,  967;

%e 14 :  0, 14, 315, 1562, 2096;

%e 15 :  0, 15, 420, 2291, 3265, 1296;

%e 16 :  0, 16, 476, 3180, 6057, 1921;

%p p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)

%p          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):

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

%p     g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-1, [i, l[]])))):

%p T:= n-> (f-> seq(coeff(f, x, i), i=0..degree(f)))(g(n$2, [])):

%p seq(T(n), n=0..20);

%t p[l_] := With[{n = Length[l]}, n!*Det[Table[With[{t = j - i + 1}, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]];

%t 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 T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];

%t Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Jul 29 2024, after _Alois P. Heinz_ *)

%Y Row sums give A265016.

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

%Y Cf. A000217, A000272, A003056, A265208.

%K nonn,tabf

%O 0,5

%A _Alois P. Heinz_, Nov 30 2015