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 1-7 of 7 results.

A330661 T(n,k) is the index within the partitions of n in canonical ordering of the k-th partition whose parts differ pairwise by at most one.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 5, 8, 9, 10, 11, 1, 5, 9, 12, 13, 14, 15, 1, 8, 13, 18, 19, 20, 21, 22, 1, 8, 19, 22, 26, 27, 28, 29, 30, 1, 13, 22, 30, 37, 38, 39, 40, 41, 42, 1, 13, 30, 41, 46, 51, 52, 53, 54, 55, 56, 1, 20, 44, 59, 62, 71, 72, 73, 74, 75, 76, 77
Offset: 1

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Author

Peter Dolland, Dec 23 2019

Keywords

Comments

For each length k in [1..n] there is exactly one such partition [p_1,...,p_k], with p_i = a+1 for i=1..j and p_i = a for i=j+1..k, where a = floor(n/k) and j = n - k * a.
If k | n, then all parts p_i are equal. A027750 lists the indices of these partitions in this triangle.
Canonical ordering is also known as graded reverse lexicographic ordering, see A080577 or link below.

Examples

			Partitions of 8 in canonical ordering begin: 8, 71, 62, 611, 53, 521, 5111, 44, 431, 422, 4211, 41111, 332, ... . The partitions whose parts differ pairwise by at most one in this list are 8, 44, 332, ... at indices 1, 8, 13, ... and this gives row 8 of this triangle.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  3,  4,  5;
  1,  3,  5,  6,  7;
  1,  5,  8,  9, 10, 11;
  1,  5,  9, 12, 13, 14, 15;
  1,  8, 13, 18, 19, 20, 21, 22;
  1,  8, 19, 22, 26, 27, 28, 29, 30;
  1, 13, 22, 30, 37, 38, 39, 40, 41, 42;
  ...

Links

Crossrefs

Cf. A000041, A063008, A027750, A080577, A216053, A238639, A238640.
T(2n,n) gives A332706.

Programs

  • Maple
    b:= proc(l) option remember; (n-> `if`(n=0, 1,
      b(subsop(1=[][], l))+g(n, l[1]-1)))(add(j, j=l))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
T:= proc(n, k) option remember; 1 + g(n$2)-
      b((q-> [q+1$r, q$k-r])(iquo(n, k, 'r')))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 19 2020
  • Mathematica
    b[l_List] := b[l] = Function[n, If[n == 0, 1, b[ReplacePart[l, 1 -> Nothing]] + g[n, l[[1]] - 1]]][Total[l]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
T[n_, k_] := T[n, k] = Module[{q, r}, {q, r} = QuotientRemainder[n, k]; 1 + g[n, n] - b[Join[Table[q + 1, {r}], Table[q, {k - r}]]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)
  • PARI
    balP(p) = p[1]-p[#p]<=1
Row(n)={v=vecsort([Vecrev(p) | p<-partitions(n)], , 4);select(i->balP(v[i]),[1..#v])}
{ for(n=1, 10, print(Row(n))) }

Formula

T(n,1) = 1.
T(n,n) = A000041(n).
T(n,k) = A000041(n) - (n - k) for k = ceiling(n/2)..n.
T(2n,2) = T(2n+1,2) = A216053(n). - Alois P. Heinz, Jan 28 2020

A332706 Index position of {2}^n within the list of partitions of 2n in canonical ordering.

Original entry on oeis.org

1, 1, 3, 8, 18, 37, 71, 128, 223, 376, 617, 991, 1563, 2423, 3704, 5589, 8333, 12293, 17959, 25996, 37318, 53153, 75153, 105535, 147249, 204201, 281563, 386128, 526795, 715191, 966437, 1300125, 1741598, 2323487, 3087701, 4087933, 5392747, 7089463, 9289053
Offset: 0

Views

Author

Alois P. Heinz, Feb 20 2020

Keywords

Comments

The canonical ordering of partitions is described in A080577.

Examples

			a(3) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Links

Crossrefs

Bisection (even part) of A058984.
Cf. A000041, A080577, A216053, A238639, A238640, A322761, A330661.

Programs

  • Maple
    a:= n-> combinat[numbpart](2*n)-n:
seq(a(n), n=0..44);
  • Mathematica
    a[n_] := PartitionsP[2n] - n;
Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Aug 20 2021, from Maple *)

Formula

a(n) = A000041(2n) - n.
a(n) = A058984(2n).
a(n) = A330661(2n,n).

A238639 Position of [n, n-1, ..., 2, 1] in Mathematica-ordered list of partitions of n(n+1)/2.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 498, 2493, 12741, 66224, 348963, 1859009, 9994196, 54155387, 295477841, 1621962199, 8951635343, 49644856801, 276540258555, 1546630084062, 8681889729354, 48900895532763, 276302483274825, 1565747892473958, 8896975706929141, 50683901455201860
Offset: 0

Views

Author

Clark Kimberling, Mar 04 2014

Keywords

Examples

			The partitions of 6 in Mathematica order are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.  The position of 321 is a(3) = 6.

Links

Crossrefs

Cf. A000217, A080577 (Mathematica ordering), A238638, A238640, A330661, A332706.

Programs

  • Maple
    g:= (n, i)-> `if`(n=0, 1, g(n-i+1, i-1)+ b(n-i, i)):
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> (m-> add(b(m-j, min(j, m-j)), j=n+1..m)+
                 g(m-n, n))(n*(n+1)/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 03 2015
  • Mathematica
    r[n_] := Table[n - k, {k, 0, n - 1}]; Flatten[Table[Position[IntegerPartitions[n (n + 1)/2], r[n]], {n, 0, 2}]]
g[n_, i_] := If[n==0, 1, g[n-i+1, i-1] + b[n-i, i]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Function[m, Sum[b[m-j, Min[j, m-j]], {j, n+1, m}] + g[m-n, n]][n(n+1)/2]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

Extensions

a(13)-a(17) from Manfred Scheucher, Jun 01 2015
a(18)-a(25) from Alois P. Heinz, Jun 02 2015

A238638 Position of n-th row of Pascal's triangle in Mathematica-ordered list of partitions of 2^n.

Original entry on oeis.org

1, 2, 4, 14, 109, 3366, 380480, 592178710, 12245355432908, 42590813279958575804, 35428820136077436448479258280, 643572551892460566707053818908283349242945, 1088540944742787295982636155758383327725184898133092177544054
Offset: 0

Views

Author

Clark Kimberling, Mar 04 2014

Keywords

Examples

			The partitions of 4 in Mathematica order are 4, 31, 22, 211, 111.  a(2) = 4 is the position of 211, which as a partition is equivalent to row 2 of Pascal's triangle:  1 2 1 (where the top row is counted as row 0).

Links

Crossrefs

Cf. A007318, A080577 (Mathematica ordering), A238639, A238640.

Programs

  • Maple
    p:= (n, k)-> binomial(n, iquo(2*n-k+1, 2)):
g:= (n, k, i)-> `if`(n=0, 1, g(n-p(k, i-1), k, i-1)
    +add(b(n-j, j), j=p(k, i-1)+1..min(n, p(k, i)))):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> (m-> add(b(m-j, min(j, m-j)), j=p(n$2)+1..m)
            +g(m-p(n$2), n$2))(2^n):
seq(a(n), n=0..10);  # Alois P. Heinz, Jun 03 2015
  • Mathematica
    r[n_] := Reverse[Sort[Table[Binomial[n, k], {k, 0, n}]]]; Flatten[Table[Position[IntegerPartitions[2^n], r[n]], {n, 0, 6}]]
(* second program: *)
$RecursionLimit = 2000;
p[n_, k_] := Binomial[n, Quotient[2*n - k + 1, 2]];
g[n_, k_, i_] := If[n == 0, 1, g[n - p[k, i - 1], k, i - 1] + Sum[b[n - j, j], {j, p[k, i - 1] + 1, Min[n, p[k, i]]}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]];
a[n_] := Function[m, Sum[b[m - j, Min[j, m - j]], {j, p[n, n] + 1, m}] + g[m - p[n, n], n, n]][2^n];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Extensions

a(7) from Manfred Scheucher, May 29 2015
a(8)-a(12) from Alois P. Heinz, Jun 03 2015

A238641 Number of partitions p of 2n-1 such that n - (number of parts of p) is a part of p.

Original entry on oeis.org

0, 0, 1, 4, 9, 21, 37, 69, 113, 187, 286, 449, 657, 976, 1397, 2003, 2788, 3902, 5323, 7284, 9789, 13144, 17405, 23052, 30142, 39379, 50967, 65842, 84368, 107954, 137126, 173893
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2014

Keywords

Examples

			a(4) counts these partitions of 7:  52, 511, 421, 331.

Links

Crossrefs

Cf. A238640, A238742.

Programs

  • Mathematica
    z = 30; g[n_] := IntegerPartitions[n]; m[p_, t_] := MemberQ[p, t];
Table[Count[g[2 n], p_ /; m[p, n - Length[p]]], {n, z}] (*A238607*)
Table[Count[g[2 n - 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238641*)
Table[Count[g[2 n + 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238742*)
p[n_, k_] := p[n, k] = If[k == 1 || n == k, 1, If[k > n, 0, p[n - 1, k - 1] + p[n - k, k]]]; q[n_, k_, e_] := q[n, k, e] = If[n - e < k - 1 , 0, If[k == 1, If[n == e, 1, 0], p[n - e, k - 1]]]; a[n_] := Sum[q[2*n - 1, u, n - u], {u, n - 1}]; Array[a,100] (* Giovanni Resta, Mar 09 2014 *)

A238742 Number of partitions p of 2n+1 such that n - (number of parts of p) is a part of p.

Original entry on oeis.org

0, 0, 1, 5, 13, 31, 59, 109, 180, 301, 461, 712, 1051, 1547, 2200, 3138, 4349, 6036, 8211, 11146, 14901, 19908, 26232, 34513, 44953, 58412, 75244, 96752, 123448, 157201, 198931, 251155
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2014

Keywords

Examples

			a(4) counts these partitions of 9:  72, 711, 621, 531, 441.

Links

Crossrefs

Cf. A238640, A238741.

Programs

  • Mathematica
    z = 30; g[n_] := IntegerPartitions[n]; m[p_, t_] := MemberQ[p, t];
Table[Count[g[2 n], p_ /; m[p, n - Length[p]]], {n, z}] (*A238607*)
Table[Count[g[2 n - 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238641*)
Table[Count[g[2 n + 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238742*)
p[n_, k_] := p[n, k] = If[k == 1 || n == k, 1, If[k > n, 0, p[n-1, k-1] + p[n-k, k]]]; q[n_, k_, e_] := q[n, k, e] = If[n-e < k-1 , 0, If[k == 1, If[n == e, 1, 0], p[n-e, k-1]]]; a[n_] := Sum[q[2*n+1, u, n-u], {u, n-1}]; Array[a, 100] (* Giovanni Resta, Mar 12 2014 *)

A332722 Index position of [2n-1, 2n-3, ..., 3, 1] within the list of partitions of n^2 in canonical ordering.

Original entry on oeis.org

1, 1, 2, 9, 74, 711, 7312, 77793, 848557, 9426039, 106218592, 1210785512, 13933358426, 161624712815, 1887635428421, 22176331059637, 261881397819259, 3106736469937751, 37006306302036790, 442425926101676831, 5306994321265281854, 63851605555921588684, 770371217568310624912
Offset: 0

Views

Author

Alois P. Heinz, Feb 20 2020

Keywords

Comments

The canonical ordering of partitions is described in A080577.

Examples

			a(3) = 9, because 531 has position 9 within the list of partitions of 3*3 in canonical ordering: 9, 81, 72, 711, 63, 621, 6111, 54, 531, ... .

Links

Crossrefs

Cf. A000041, A000290, A005408, A080577, A238639, A238640, A322761.

Programs

  • Maple
    b:= proc(n, i) option remember;
     `if`(n=0, 1, b(n-i, i-2)+g(n, i-1))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(n^2$2)-b(n^2, 2*n-1)+1:
seq(a(n), n=0..23);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i - 2] + g[n, i - 1]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[n^2, n^2] - b[n^2, 2n - 1] + 1;
a /@ Range[0, 23] (* Jean-François Alcover, May 10 2020, after Maple *)
Showing 1-7 of 7 results.

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