A135010 -id:A135010 - cp's OEIS frontend

A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

1, 2, 0, 3, 1, 0, 5, 0, 0, 0, 7, 2, 1, 0, 0, 11, 0, 0, 0, 0, 0, 15, 3, 0, 1, 0, 0, 0, 22, 0, 2, 0, 0, 0, 0, 0, 30, 5, 0, 0, 1, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Nov 20 2009

In the square array, note that the column k starts with k-1 zeros. Then list each partition number of positive integers followed by k-1 zeros. See A000041, which is the main entry for this sequence.

			The array, A(n, k), begins:
   n | k = 1   2   3   4   5   6   7   8   9  10  11  12
  ---+--------------------------------------------------
   1 |     1   0   0   0   0   0   0   0   0   0   0   0
   2 |     2   1   0   0   0   0   0   0   0   0   0   0
   3 |     3   0   1   0   0   0   0   0   0   0   0   0
   4 |     5   2   0   1   0   0   0   0   0   0   0   0
   5 |     7   0   0   0   1   0   0   0   0   0   0   0
   6 |    11   3   2   0   0   1   0   0   0   0   0   0
   7 |    15   0   0   0   0   0   1   0   0   0   0   0
   8 |    22   5   0   2   0   0   0   1   0   0   0   0
   9 |    30   0   3   0   0   0   0   0   1   0   0   0
  10 |    42   7   0   0   2   0   0   0   0   1   0   0
  11 |    56   0   0   0   0   0   0   0   0   0   1   0
  12 |    77  11   5   3   0   2   0   0   0   0   0   1
  ...
Antidiagonal triangle, T(n,k), begins as:
   1;
   2, 0;
   3, 1, 0;
   5, 0, 0, 0;
   7, 2, 1, 0, 0;
  11, 0, 0, 0, 0, 0;
  15, 3, 0, 1, 0, 0, 0;
  22, 0, 2, 0, 0, 0, 0, 0;
  30, 5, 0, 0, 1, 0, 0, 0, 0;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0;

G. C. Greubel, Antidiagonals n = 1..50, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000007, A000041, A035377, A035444, A135010, A138121.

Cf. A168016, A168017, A168018, A168019, A168021. [From Omar E. Pol, Nov 23 2009]

T[n_, k_]:= If[IntegerQ[(n-k+1)/k], PartitionsP[(n-k+1)/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def A168020(n,k): return number_of_partitions((n-k+1)/k) if ((n-k+1)%k)==0 else 0
flatten([[A168020(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

A(n, k) = A000041(n/k) if k divides n, otherwise A(n, k) = 0 (array).
A(n, 1) = A(n*k, k) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(n, k) = A000041((n-k+1)/k) if k divides (n-k+1), otherwise T(n, k) = 0 (triangle).
T(n, 1) = A000041(n).
T(2*n, n) = 2*A000007(n-1), n >= 1. (End)

Edited by Omar E. Pol, Nov 21 2009
Edited by Charles R Greathouse IV, Mar 23 2010
Edited by Max Alekseyev, May 07 2010

A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Dec 01 2010

Triangle read by rows in which row n lists the smallest parts of all partitions of n in the order produced by the shell model of partitions of A138121.
Also, row n lists the "filler parts" of all partition of n. For more information see A182699.
Row n has length A000041(n). Row sums give A046746. Column 1 gives A001477. The last A000041(n-1) terms of row n are ones, n >= 1.

			For row 10, see the illustration of the link.
Triangle begins:
  0,
  1,
  2,1,
  3,1,1,
  4,2,1,1,1,
  5,2,1,1,1,1,1,
  6,3,2,2,1,1,1,1,1,1,1,
  7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
  8,4,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  9,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ...

Omar E. Pol, Illustration of the smallest parts of all partitions of 10 (colored blue) in the shell model of partitions

Cf. A026807, A135010, A138121, A141285, A182699, A182709.

Mirror of triangle A196931.

Name simplified and more terms from Omar E. Pol, Oct 21 2011

A182731 Odd-indexed rows of triangle A141285.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 4, 7, 3, 5, 4, 7, 3, 6, 5, 9, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11

Offset: 1

Omar E. Pol, Nov 28 2010

			Triangle begins:
1,
3,
3, 5,
3, 5, 4, 7,
3, 5, 4, 7, 3, 6, 5, 9,
3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11,

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A135010, A138121, A141285, A182730, A182732.

Rows converge to A182733.

A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

Original entry on oeis.org

0, -1, 1, -2, 0, 2, -3, -1, 1, 0, 3, -4, -2, 0, -1, 2, 1, 4, -5, -3, -1, -2, 1, 0, 3, -1, 2, 1, 5, -6, -4, -2, -3, 0, -1, 2, -2, 1, 0, 4, 0, 3, 2, 6, -7, -5, -3, -4, -1, -2, 1, -3, 0, -1, 3, -1, 2, 1, 5, -2, 1, 0, 4, 3, 2, 7, -8, -6, -4, -5, -2, -3, 0, -4, -1

Offset: 1

Omar E. Pol, Dec 10 2011

Row n has length A000041(n). The sum of row n is equal to zero.

			Written as a triangle:
  0;
  -1,1;
  -2,0,2;
  -3,-1,1,0,3;
  -4,-2,0,-1,2,1,4;
  -5,-3,-1,-2,1,0,3,-1,2,1,5;
  -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
  -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;

Alois P. Heinz, Rows n = 1..26, flattened

Cf. A105805, A135010, A138121, A194546, A194548, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      l:= NULL; b(n, 1, 0); l
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 22 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] := If [n0==0, l = Append[l, i-t], If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]]; l = {}; b[n, 1, 0]; l];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A194546(n) - A193173(n).

More terms from Alois P. Heinz, Dec 22 2011

A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 6, 5, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  0;
  1;
  1;
  2,1;
  2,1;
  3,2,2,1;
  3,2,2,1;
  4,3,3,2,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,3,3,2,3,2,2,2,1;
  5,4,4,3,4,3,3,2,3,3,2,2,2,1;
  6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
  6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;

Alois P. Heinz, Rows n = 1..33, flattened
Tilman Piesk, Table for A194602, showing the non-one addends.

Row sums give A138135. Row n has length A187219(n).

Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Dec 19 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0==0, l = Append[l, t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 15}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 19 2011

A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, -1, 2, 1, 5, 0, 3, 2, 6, -2, 1, 0, 4, 3, 2, 7, -1, 2, 1, 5, 0, 4, 3, 8, -3, 0, -1, 3, 2, 1, 6, 1, 5, 4, 3, 9, -2, 1, 0, 4, -1, 3, 2, 7, 2, 1, 6, 5, 4, 10, -4, -1, -2, 2, 1, 0, 5, 0, 4, 3, 2, 8, -1, 3, 2, 7, 1, 6, 5, 4, 11, -3, 0, -1, 3, -2

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  1;
  1;
  2;
  0,3;
  1,4;
  -1,2,1,5;
  0,3,2,6;
  -2,1,0,4,3,2,7;
  -1,2,1,5,0,4,3,8;
  -3,0,-1,3,2,1,6,1,5,4,3,9;
  -2,1,0,4,-1,3,2,7,2,1,6,5,4,10;
  -4,-1,-2,2,1,0,5,0,4,3,2,8,-1,3,2,7,1,6,5,4,11;

Alois P. Heinz, Rows n = 1..33, flattened

The sum of row n is A000041(n-1). Row n has length A187219(n).

Cf. A002865, A135010, A138121, A194546, A194547, A194548.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 1 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..13); # Alois P. Heinz, Dec 20 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0 == 0, l = Append[l, i-t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {1}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 13}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A141285(n) - A194548(n).

More terms from Alois P. Heinz, Dec 20 2011

A194702 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).

Original entry on oeis.org

2, 0, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 2. For further information see A182703 and A135010.

			Triangle begins:
2,
0, 2,
1, 0, 1,
0, 1, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 0, 1,
...
For k = 1 and  m = 1; T(1,1) = 2 because there are two parts of size 1 in the last section of the set of partitions of 3, since 2 + m = 3, so a(1) = 2. For k = 2 and m = 1; T(2,1) = 0 because there are no parts of size 2 in the last section of the set of partitions of 3, since 2 + m = 3, so a(2) = 0.

Always the sum of row k = p(2) = A000041(n) = 2.
The first (0-10) members of this family of triangles are A023531, A129186, this sequence, A194703-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

T(k,m) = A182703(2+m,k), with T(k,m) = 0 if k > 2+m.

T(k,m) = A194812(2+m,k).

A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

Original entry on oeis.org

42, 15, 27, 10, 14, 18, 5, 10, 10, 17, 4, 5, 8, 10, 15, 2, 5, 4, 8, 9, 14, 2, 2, 4, 5, 7, 9, 13, 1, 2, 2, 4, 4, 8, 8, 13, 1, 1, 2, 2, 4, 4, 7, 9, 12, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of row k is also the number of partitions of 10. For further information see A182703 and A135010.

			Triangle begins:
  42;
  15, 27;
  10, 14, 18;
   5, 10, 10, 17;
   4,  5,  8, 10, 15;
   2,  5,  4,  8,  9, 14;
   2,  2,  4,  5,  7,  9, 13;
   1,  2,  2,  4,  4,  8,  8, 13;
   1,  1,  2,  2,  4,  4,  7,  9, 12;
   0,  1,  1,  2,  2,  4,  4,  7,  8, 13;
   1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
  ...
For k = 1 and m = 1; T(1,1) = 42 because there are 42 parts of size 1 in the last section of the set of partitions of 11, since 10 + m = 11, so a(1) = 42. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 11, since 10 + m = 11, so a(2) = 15.

Always the sum of row k = p(10) = A000041(10) = 42.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194709, this sequence.
Cf. A002865, A135010, A138121, A194812.

T(k,m) = A182703(10+m,k), with T(k,m) = 0 if k > 10+m.
T(k,m) = A194812(10+m,k).
Beginning with row k=11 each row starts with (k-11) 0's and ends with the subsequence 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, the initial terms of A002865. - Alois P. Heinz, Feb 15 2012

A196931 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Oct 21 2011

If n >= 1, row n lists the smallest parts of every partition of n in the order produced by the shell model of partitions of A135010, hence row n lists the parts of the last section of the set of partitions of n, except the emergent parts (See A182699).

Row n has length A000041(n). Row sums give A046746. Right border of triangle gives A001477. Row n starts with A000041(n-1) ones, n >= 1.

			Written as a triangle:
  0,
  1,
  1,2,
  1,1,3,
  1,1,1,2,4,
  1,1,1,1,1,2,5,
  1,1,1,1,1,1,1,2,2,3,6
  1,1,1,1,1,1,1,1,1,1,1,2,2,3,7,
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,8,
  ...

Mirror of triangle A182715.

Cf. A000041, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196930.

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 2, 5, 5, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 6, 1, 3, 3, 1, 4, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 5, 1, 1, 3, 2, 2, 5, 2, 4, 3, 7

Offset: 1

Omar E. Pol, Aug 14 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First we copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.

			A table of partitions.
---------------------------------------------------------
.              Expanded       Geometric  Side view of the
Partitions     version        model      folded table
---------------------------------------------------------
1;             1;             |*|                /
---------------------------------------------------------
1,1;           1,1;           |o|*|              \
2;             . 2;           |* *|               \
---------------------------------------------------------
2,1;           . 2,1;         |o o|*|             /
1,1,1;         1,1,1;         |o|o|*|            /
3;             . . 3;         |* * *|           /
---------------------------------------------------------
3,1;           . . 3,1;       |o o o|*|         \
1,1,1,1;       1,1,1,1;       |o|o|o|*|          \
2,1,1;         . 2,1,1;       |o o|o|*|           \
2,2;           . 2,. 2;       |* *|* *|            \
4;             . . . 4;       |* * * *|             \
---------------------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|           /
2,2,1;         . 2,. 2,1;     |o o|o o|*|          /
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|         /
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|        /
3,1,1;         . . 3,1,1;     |o o o|o|*|       /
3,2;           . . 3,. 2;     |* * *|* *|      /
5;             . . . . 5;     |* * * * *|     /
---------------------------------------------------------
5,1;           . . . . 5,1;   |o o o o o|*|   \
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|    \
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|     \
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|      \
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|       \
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|        \
4,1,1;         . . . 4,1,1;   |o o o o|o|*|         \
2,2,2;         . 2, .2,. 2;   |* *|* *|* *|          \
4,2;           . . . 4,. 2;   |* * * *|* *|           \
3,3;           . . 3,. . 3;   |* * *|* * *|            \
6;             . . . . . 6;   |* * * * * *|             \
---------------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Omar E. Pol, Illustration of initial terms

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

cp's OEIS Frontend

A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

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A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

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A182731 Odd-indexed rows of triangle A141285.

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A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

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A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

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A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

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A194702 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).

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A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

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