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 1-6 of 6 results.

A265018 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), n>=0, floor(sqrt(2n)+1/2)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 4, 8, 7, 5, 16, 12, 9, 6, 40, 16, 11, 7, 34, 55, 20, 13, 8, 50, 73, 70, 24, 15, 9, 125, 132, 96, 85, 28, 17, 10, 281, 212, 119, 100, 32, 19, 11, 351, 469, 267, 142, 115, 36, 21, 12, 307, 642, 644, 322, 165, 130, 40, 23, 13
Offset: 0

Views

Author

Alois P. Heinz, Nov 30 2015

Keywords

Examples

			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;

Links

Crossrefs

Row sums give A265016.
Column sums give A265130.
Cf. A000217, A000272, A002024, A265019 (the same read by columns).

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=ldegree(f)..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);

Formula

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

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

Views

Author

Alois P. Heinz, Nov 30 2015

Keywords

Examples

			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;

Links

Crossrefs

Row sums give A265016.
Column sums give A265130.
Cf. A000217, A000272, A265018 (the same read by rows).

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[]])))):
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);
  • Mathematica
    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 *)

Formula

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

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 *)

A265007 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n.

Original entry on oeis.org

1, 1, 3, 7, 18, 40, 97, 216, 499, 1112, 2502, 5503, 12197, 26582, 58088, 125619, 271713, 583228, 1251115, 2668651, 5685053, 12059993, 25544291, 53926003, 113666195, 238946232, 501546514, 1050430420, 2196869731, 4586021745, 9560876381, 19900839742, 41373446190
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2015

Keywords

Examples

			The number of lambda-parking functions induced by the partitions of 4:
1 by [1,1,1,1]: [1,1,1,1],
4 by [1,1,2]: [1,1,1], [1,1,2], [1,2,1], [2,1,1],
4 by [2,2]: [1,1], [1,2], [2,1], [2,2],
5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
4 by [4]: [1], [2], [3], [4].
a(4) = 1 + 4 + 4 + 5 + 4 = 18.

Links

Crossrefs

Cf. A000041, A255047, A265016.

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`(n=0 or i=1, p([1$n, l[]]), g(n, i-1, l)
               +`if`(i>n, 0, g(n-i, i, [i, l[]]))):
a:= n-> g(n$2, []):
seq(a(n), n=0..20);
  • Mathematica
    p[l_] := With[{n = Length[l]}, n! Det[Table[With[{t = j - i + 1},
     If[t < 0, 0, l[[i]]^t/t!]], {i, n}, {j, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, p[Join[
     Table[1, {n}], l]], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i, Prepend[l, i]]]];
a[n_] := If[n == 0, 1, g[n, n, {}]];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Aug 22 2021, after Alois P. Heinz *)

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).

A265017 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with smallest part k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 5, 0, 0, 4, 0, 7, 8, 0, 0, 5, 0, 25, 12, 0, 0, 0, 6, 0, 36, 16, 15, 0, 0, 0, 7, 0, 81, 20, 21, 0, 0, 0, 0, 8, 0, 107, 74, 27, 24, 0, 0, 0, 0, 9, 0, 316, 102, 33, 32, 0, 0, 0, 0, 0, 10, 0, 427, 222, 39, 40, 35, 0, 0, 0, 0, 0, 11
Offset: 0

Views

Author

Alois P. Heinz, Nov 30 2015

Keywords

Examples

			Triangle T(n,k) begins:
00 :  1;
01 :  0,   1;
02 :  0,   0,   2;
03 :  0,   3,   0,   3;
04 :  0,   5,   0,   0,  4;
05 :  0,   7,   8,   0,  0,  5;
06 :  0,  25,  12,   0,  0,  0, 6;
07 :  0,  36,  16,  15,  0,  0, 0, 7;
08 :  0,  81,  20,  21,  0,  0, 0, 0, 8;
09 :  0, 107,  74,  27, 24,  0, 0, 0, 0, 9;
10 :  0, 316, 102,  33, 32,  0, 0, 0, 0, 0, 10;
11 :  0, 427, 222,  39, 40, 35, 0, 0, 0, 0,  0, 11;
12 :  0, 869, 286, 153, 48, 45, 0, 0, 0, 0,  0,  0, 12;

Links

Crossrefs

Row sums give A265016.
Column k=0 gives A000007.
Main diagonal gives A028310, first lower diagonal is A000004.
T(2n+1,n) gives A005563.
T(2n+2,n) gives A028347(n+2).
T(2n+3,n) gives A028560.

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..n))(g(n$2, [])):
seq(T(n), n=0..16);
  • Mathematica
    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]*x^
     If[l == {}, 0, l[[1]]], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)
Showing 1-6 of 6 results.

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