A026836 -id:A026836 - cp's OEIS frontend

A026839 a(n) = MAX{T(n,k) for k=1,2,...,n}, T given in A026836.

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 20, 22, 25, 28, 32, 36, 40, 45, 51, 57, 64, 71, 79, 89, 99, 110, 122, 137, 152, 168, 186, 208, 230, 253, 280, 311, 343, 378, 416, 461, 507, 558, 613, 675

Offset: 1

Clark Kimberling

nonn

A072575 Triangle T(n,k) of number of compositions (ordered partitions) of n into distinct parts where largest part is exactly k, 1<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 6, 2, 2, 1, 0, 0, 0, 8, 2, 2, 1, 0, 0, 0, 6, 8, 2, 2, 1, 0, 0, 0, 6, 8, 8, 2, 2, 1, 0, 0, 0, 24, 12, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 30, 36, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 36, 38, 14, 8, 8, 2, 2, 1, 0, 0, 0, 0, 24, 54, 42, 38, 14, 8, 8, 2, 2, 1

Offset: 1

Henry Bottomley, Jun 21 2002

			Rows start:
  1;
  0, 1;
  0, 2, 1;
  0, 0, 2, 1;
  0, 0, 2, 2, 1;
  0, 0, 6, 2, 2, 1;
  0, 0, 0, 8, 2, 2, 1;
  0, 0, 0, 6, 8, 2, 2, 1;
  ...
T(7,4)=8 since 7 can be written as 4+3 =4+2+1 =4+1+2 =3+4 =2+4+1 =2+1+4 =1+4+2 =1+2+4.

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for sequences related to compositions

Cf. A026836, A072574. Row sums are A032020. Column sums appear to be A001339 (offset). Starting terms of columns tend towards A072576 as k increases.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, [][], zip((x, y)->x+y, [b(n, i-1)],
      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
    end:
T:= proc(n, k) local l; l:= [b(n-k, k-1)];
       add(l[i]*(i)!, i=1..nops(l))
    end:
seq(seq(T(n, k), k=1..n), n=1..20);  # Alois P. Heinz, Nov 20 2012

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], If[i>n, {}, Join[{0}, b[n-i, i-1]]]}]]]; T[n_, k_] := Module[{l}, l = b[n-k, k-1]; Sum[l[[i]]*i!, {i, 1, Length[l]}]]; Table[Table [T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 31 2014, after Alois P. Heinz *)

A059607 As an upper right triangle, number of distinct partitions of n where the highest part is k (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1

Offset: 0

Henry Bottomley, Jan 30 2001

			Rows are {1,0,0,0,...}, {1,0,0,0,...}, {1,1,0,0,...}, {1,1,1,1,...}, {1,1,1,2,...} etc. T(7,4)=2 since 7 can be written as 4+3 or 4+2+1. T(12,6)=3 since 12 can be written as 6+5+1 or 6+4+2 or 6+3+2+1.

As upper right triangle, row sum is A011782, column sum is A000009, column maximum is A025591 (offset), row maximum is A026839 (offset). Cf. A026836 for this triangle starting at (1, 1) rather than (0, 0).

t[n_?Positive, k_] := t[n, k] = Sum[t[n-k, j], {j, 0, k-1}]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ] = 0; Table[ t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover, Sep 11 2012 *)

T(n, k) =sum_j[T(n-k, j)] for k>j with T(0, 0)=1

A070936 Square array read by antidiagonals: T(n,k) = number of partitions of n into distinct parts, each no more than k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 1, 0, 0, 0, 1, 1, 1, 2, 2, 3, 2, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 2, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 3, 1, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 0, 0, 0, 0

Offset: 0

Henry Bottomley, May 12 2002

			Rows start
1,1,1,1,1,...;
0,1,1,1,1,...;
0,0,1,1,1,...;
0,0,1,2,2,...;
0,0,0,1,2,...; etc.
T(10,5)=3 since 10 can be partitioned 3 ways as 5+4+1=5+3+2=4+3+2+1 with each part less than or equal to 5.

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A008284, A060016. With some imagination, this is the transpose of A026836 and A053632. Column sums are 2^k=A000079(k). Column maximum is A025591(k), which appears A070936(k) times in the column.

T(n, k) =T(n-1, k)+T(n-1, k-n) (with T(0, 0)=1) =A053632(k, n) =A026836(n+k+1, k+1) =sum_{0<=j<=k}A026836(n, j). For k>=n, T(n, k)=T(n, n)=A000009(n).

A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3

Offset: 1

Henry Bottomley, Feb 01 2001

Weakly unimodal means nondecreasing then nonincreasing.

			Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
As an upper right triangle:
  1,  1,  2,  4,  8, 15, ...,
      1,  1,  2,  3,  5,  8, ...,
          1,  1,  2,  3,  5,  7, ...,
              ...
As a left downward triangle, it starts:
   1;
   1, 1;
   2, 1, 1;
   4, 2, 1, 1;
   8, 3, 2, 1, 1;
  15, 5, 3, 2, 1, 1;
  27, 8, 5, 3, 2, 1, 1;
  ...
T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.

Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.

T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.

A026840 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts <= k, for k=1,2,...n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 3, 4, 0, 0, 0, 2, 3, 4, 5, 0, 0, 0, 1, 3, 4, 5, 6, 0, 0, 0, 1, 3, 5, 6, 7, 8, 0, 0, 0, 1, 3, 5, 7, 8, 9, 10, 0, 0, 0, 0, 2, 5, 7, 9, 10, 11, 12, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15, 0, 0, 0, 0, 1, 4, 8, 11

Offset: 1

Clark Kimberling

Cf. A026836.

Differs from A079126 in having fewer zeros at the ends of the rows.

cp's OEIS Frontend

A026839 a(n) = MAX{T(n,k) for k=1,2,...,n}, T given in A026836.

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A072575 Triangle T(n,k) of number of compositions (ordered partitions) of n into distinct parts where largest part is exactly k, 1<=k<=n.

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A059607 As an upper right triangle, number of distinct partitions of n where the highest part is k (0<=k<=n).

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A070936 Square array read by antidiagonals: T(n,k) = number of partitions of n into distinct parts, each no more than k.

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A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

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A026840 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts <= k, for k=1,2,...n.

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