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.

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A265146 Triangle T(n,k) in which n-th row lists the parts i_1=1, 1<=k<=A001222(n).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 2, 3, 1, 4, 5, 1, 2, 4, 6, 1, 5, 2, 4, 1, 2, 3, 4, 7, 1, 3, 4, 8, 1, 2, 5, 2, 5, 1, 6, 9, 1, 2, 3, 5, 3, 4, 1, 7, 2, 3, 4, 1, 2, 6, 10, 1, 3, 5, 11, 1, 2, 3, 4, 5, 2, 6, 1, 8, 3, 5, 1, 2, 4, 5, 12, 1, 9, 2, 7, 1, 2, 3, 6, 13, 1
Offset: 1

Views

Author

Alois P. Heinz, Dec 02 2015

Keywords

Comments

A strict partition is a partition into distinct parts.
Row n=1 contains the parts of the empty partition, so it is empty.

Examples

			n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].
Triangle T(n,k) begins:
01 :  ;
02 :  1;
03 :  2;
04 :  1, 2;
05 :  3;
06 :  1, 3;
07 :  4;
08 :  1, 2, 3;
09 :  2, 3;
10 :  1, 4;
11 :  5;
12 :  1, 2, 4;
13 :  6;
14 :  1, 5;
15 :  2, 4;
16 :  1, 2, 3, 4;

Links

Crossrefs

Column k=1 gives A055396 (for n>1).
Last terms of rows give A252464 (for n>1).
Row sums give A266475.
Cf. A000009, A000040, A001222, A112798, A246688, A265145.

Programs

  • Maple
    T:= n-> ((l-> seq(l[j]+j-1, j=1..nops(l)))(sort([seq(
       numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(T(n), n=1..100);
  • Mathematica
    T[n_] := Function[l, Table[l[[j]]+j-1, {j, 1, Length[l]}]][Sort[ Flatten[ Table[ Array[ PrimePi[i[[1]]]&, i[[2]]], {i, FactorInteger[n]}]]]];
Table[T[n], {n, 1, 100}] // Flatten // Rest (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

T(prime(n),1) = 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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Dec 04 2015

Keywords

Comments

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

Examples

			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;

Links

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.

Programs

  • Maple
    b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
      `if`(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[nJean-François Alcover, Feb 02 2017, translated from Maple *)

Formula

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

A265144 Number of lambda-parking functions of the unique partition lambda with encoding n = Product_{i:lambda} prime(i).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 4, 1, 4, 5, 5, 4, 6, 7, 8, 1, 7, 7, 8, 7, 12, 9, 9, 5, 9, 11, 8, 10, 10, 16, 11, 1, 16, 13, 15, 11, 12, 15, 20, 9, 13, 25, 14, 13, 20, 17, 15, 6, 16, 19, 24, 16, 16, 15, 21, 13, 28, 19, 17, 27, 18, 21, 32, 1, 27, 34, 19, 19, 32, 34, 20, 16
Offset: 1

Views

Author

Alois P. Heinz, Dec 02 2015

Keywords

Examples

			n = 18 = 2*3*3 = prime(1)*prime(2)*prime(2) encodes partition [1,2,2] having seven lambda-parking functions: [1,1,1], [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], thus a(18) = 7.

Links

Crossrefs

Cf. A000040, A000041, A215366, A265145.

Programs

  • Maple
    with(numtheory):
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(sort([seq(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[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 *)
Showing 1-3 of 3 results.

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