cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A265145 Number of lambda-parking functions of the unique strict partition lambda with parts i_1

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 4, 16, 8, 7, 5, 25, 6, 9, 12, 125, 7, 34, 8, 34, 16, 11, 9, 189, 15, 13, 50, 43, 10, 49, 11, 1296, 20, 15, 21, 243, 12, 17, 24, 253, 13, 64, 14, 52, 74, 19, 15, 1921, 24, 58, 28, 61, 16, 307, 27, 317, 32, 21, 17, 343, 18, 23, 98, 16807, 33
Offset: 1

Views

Author

Alois P. Heinz, Dec 02 2015

Keywords

Comments

A strict partition is a partition into distinct parts.

Examples

			n = 10 = 2*5 = prime(1)*prime(3) encodes strict partition [1,4] having seven lambda-parking functions: [1,1], [1,2], [2,1], [1,3], [3,1], [1,4], [4,1], thus a(10) = 7.

Links

Crossrefs

Cf. A000009, A000040, A265144, A265146, A265208.

Programs

  • Maple
    p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
a:= n-> p((l-> [seq(l[j]+j-1, j=1..nops(l))])(sort([seq(
         numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(a(n), n=1..100);
  • Mathematica
    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]];
a[n_] := If[n==1, 1, p[Function[l, Flatten[Table[l[[j]]+j-1,
     {j, 1, Length[l]}]]][Sort[Flatten[Table[Table[PrimePi[
     i[[1]]], {i[[2]]}], {i, FactorInteger[n]}]]]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 21 2021, after Alois P. Heinz *)

A265202 Total number of lambda-parking functions induced by all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 2, 6, 9, 15, 36, 53, 78, 119, 286, 401, 591, 829, 1232, 2910, 4084, 5789, 8070, 11281, 15823, 37747, 51622, 72919, 98986, 136600, 181648, 254638, 586891, 799841, 1110303, 1495279, 2018749, 2657612, 3552560, 4738775, 10857521, 14560375, 20061359, 26603227
Offset: 0

Views

Author

Alois P. Heinz, Dec 04 2015

Keywords

Examples

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 2: [1], [2].
a(3) = 6: [1], [2], [3], [1,1], [1,2], [2,1].
a(4) = 9: [1], [2], [3], [4], [1,1], [1,2], [1,3], [2,1], [3,1].
a(5) = 15: [1], [2], [3], [4], [5], [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1].
a(6) = 36: [1], [2], [3], [4], [5], [6], [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], [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].

Links

Crossrefs

Row sums of A265208.
Cf. A000009, A255047, A265016.

Programs

  • Maple
    b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!)+
      `if`(n `if`(n=0, 1, b(0$2, n, 1$2)):
seq(a(n), n=0..50);
  • Mathematica
    b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!] + If[nJean-François Alcover, Feb 02 2017, translated from Maple *)

A265020 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.

Original entry on oeis.org

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, 142, 0, 10, 110, 248, 125, 0, 11, 165, 409, 189, 0, 12, 195, 710, 496, 0, 13, 273, 1033, 967, 0, 14, 315, 1562, 2096, 0, 15, 420, 2291, 3265, 1296, 0, 16, 476, 3180
Offset: 0

Views

Author

Alois P. Heinz, Nov 30 2015

Keywords

Comments

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

Examples

			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.
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  15;
06 :  0,  6,  21,   16;
07 :  0,  7,  42,   25;
08 :  0,  8,  54,   68;
09 :  0,  9,  90,  142;
10 :  0, 10, 110,  248,  125;
11 :  0, 11, 165,  409,  189;
12 :  0, 12, 195,  710,  496;
13 :  0, 13, 273, 1033,  967;
14 :  0, 14, 315, 1562, 2096;
15 :  0, 15, 420, 2291, 3265, 1296;
16 :  0, 16, 476, 3180, 6057, 1921;

Links

Crossrefs

Row sums give A265016.
Columns k=0-1 give: A000007, A000027.
Cf. A000217, A000272, A003056, A265208.

Programs

  • Maple
    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[]])))):
T:= n-> (f-> seq(coeff(f, x, i), i=0..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);
  • Mathematica
    p[l_] := With[{n = Length[l]}, n!*Det[Table[With[{t = j - i + 1}, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]];
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[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jul 29 2024, after Alois P. Heinz *)

Formula

T(A000217(n),n) = A000272(n+1).
Showing 1-3 of 3 results.

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