A265016
Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.
Original entry on oeis.org
1, 1, 2, 6, 9, 20, 43, 74, 130, 241, 493, 774, 1413, 2286, 3987, 7287, 11650, 19235, 31581, 50852, 80867, 141615, 214538, 349179, 541603, 859759, 1303221, 2054700, 3277493, 4960397, 7652897, 11662457, 17703655, 26603187, 40043433, 59384901, 92234897, 134538472
Offset: 0
The number of lambda-parking functions induced by the partitions of 4 into distinct parts:
5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
4 by [4]: [1], [2], [3], [4].
a(4) = 5 + 4 = 9.
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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)/2n, 0, g(n-i, i-1, [i, l[]])))):
a:= n-> g(n$2, []):
seq(a(n), n=0..35);
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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],
g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
a[n_] := If[n == 0, 1, g[n, n, {}]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)
A265019
Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.
Original entry on oeis.org
1, 1, 2, 3, 3, 5, 8, 16, 4, 7, 12, 40, 34, 50, 125, 5, 9, 16, 55, 73, 132, 281, 351, 307, 432, 1296, 6, 11, 20, 70, 96, 212, 469, 642, 1020, 1361, 3294, 3305, 3910, 3506, 4802, 16807, 7, 13, 24, 85, 119, 267, 644, 959, 1567, 2686, 5570, 7109, 11890, 13234
Offset: 0
Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 2;
03 : 3, 3;
04 : 5, 4;
05 : 8, 7, 5;
06 : 16, 12, 9, 6;
07 : 40, 16, 11, 7;
08 : 34, 55, 20, 13, 8;
09 : 50, 73, 70, 24, 15, 9;
10 : 125, 132, 96, 85, 28, 17, 10;
11 : 281, 212, 119, 100, 32, 19, 11;
12 : 351, 469, 267, 142, 115, 36, 21, 12;
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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)/2n, 0, g(n-i, i-1, [i, l[]])))):
b:= proc(n) option remember; g(n$2, []) end:
T:= k-> seq(coeff(b(n), x, k), n=k..k*(k+1)/2):
seq(T(k), k=0..8);
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p[l_] := With[{n = Length[l]}, n!*Det[Table[t = j-i+1; If[t<0, 0, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[i*(i+1)/2n, 0, g[n-i, i-1, Join[{i}, l]]]]]; b[n_] := b[n] = g[n, n, {}]; T[0] = {1}; T[k_] := Table[Coefficient[b[n], x, k], {n, k, k*(k+1)/2}]; Table[T[k], {k, 0, 8}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)
A265130
Total sum of number of lambda-parking functions, where lambda ranges over all partitions of k into distinct parts with largest part n and n<=k<=n*(n+1)/2.
Original entry on oeis.org
1, 1, 5, 32, 272, 2957, 39531, 629806, 11673074, 247028567, 5881190801, 155651692748, 4534744862052, 144246963009697, 4975152075900887, 184958685188293274, 7373625038400716198, 313817002976857310507, 14201832585602869616349, 681022860320979979626232
Offset: 0
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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)/2n, 0, g(n-i, i-1, [i, l[]])))):
a:= n-> `if`(n=0, 1, add(g(k-n, n-1, [n]), k=n..n*(n+1)/2)):
seq(a(n), n=0..10);
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p[l_] := Function[n, n!*Det[Table[Function [t,
If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]][Length[l]];
g[n_, i_, l_] := If[i(i+1)/2 < n, 0,
If[n == 0, p[l], g[n, i - 1, l] +
If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
a[n_] := If[n == 0, 1, Sum[g[k - n, n - 1, {n}], {k, n, n(n+1)/2}]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Aug 22 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.