A239512 -id:A239512 - cp's OEIS frontend

A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

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

Offset: 1

Omar E. Pol, Apr 05 2018

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nonincreasing order. In other words: row n lists in nonincreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

			Triangle begins:
  1;
  2,1,1;
  3,2,1,1,1,1;
  4,3,2,2,2,1,1,1,1,1,1,1;
  5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1;
  6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There is only one 4, only one 3, three 2's and seven 1's, so the 4th row of this triangle is [4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nonincreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Paolo Xausa, Table of n, a(n) for n = 1..9687, (rows 1..18 of triangle, flattened).

Both column 1 and 2 are A000027.
Row n has length A006128(n).
The sum of row n is A066186(n).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
First differs from A036037, A080577, A181317, A237982 and A239512 at a(13) = T(4,3).
Cf. A302247 (mirror).
Cf. A000041, A027550, A176206, A221529, A336812, A338156.

nrows=10;Array[ReverseSort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)

row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list), , 4); \\ Michel Marcus, Jun 16 2022

A239001 Irregular triangular array read by rows: row n gives a list of the partitions of n into Fibonacci numbers.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 3, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 5, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 1, 1, 3, 1, 1

Offset: 1

Clark Kimberling, Mar 08 2014

The number of partitions represented in row n is A003107(n).

The parts of a partition are nonincreasing and the order of the partitions is anti-lexicographic. As parts one uses A000045(n), n >= 2. - Wolfdieter Lang, Mar 17 2014

			1
2 1 1
3 2 1 1 1 1
3 1 2 2 2 1 1 1 1 1 1
5 3 2 3 1 1 2 2 1 2 1 1 1 1 1 1 1 1
Row 5 represents these six partitions: 5, 32, 311, 221, 2111, 11111.
From _Wolfdieter Lang_, Mar 17 2014: (Start)
The array with separated partitions begins:
n\k   1      2      3        4        5          6          7            8             9             10 ...
1:    1
2:    2    1,1
3:    3    2,1  1,1,1
4:  3,1    2,2  2,1,1  1,1,1,1
5:    5    3,2  3,1,1    2,2,1  2,1,1,1  1,1,1,1,1
6:  5,1    3,3  3,2,1  3,1,1,1    2,2,2    2,2,1,1  2,1,1,1,1  1,1,1,1,1,1
7:  5,2  5,1,1  3,3,1    3,2,2  3,2,1,1  3,1,1,1,1    2,2,2,1    2,2,1,1,1   2,1,1,1,1,1  1,1,1,1,1,1,1
...
Row n=8: 8  5,3  5,2,1  5,1,1,1  3,3,2  3,3,1,1  3,2,2,1  3,2,1,1,1  3,1,1,1,1,1   2,2,2,2   2,2,2,1,1
  2,2,1,1,1,1  2,1,1,1,1,1,1  1,1,1,1,1,1,1,1;
Row n=9  8,1  5,3,1  5,2,2   5,2,1,1   5,1,1,1,1  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   3,1,1,1,1,1,1  2,2,2,2,1  2,2,2,1,1,1  2,2,1,1,1,1,1   2,1,1,1,1,1,1,1   1,1,1,1,1,1,1,1,1;
Row n=10: 8,2  8,1,1   5,5   5,3,2  5,3,1,1  5,2,2,1  5,2,1,1,1  5,1,1,1,1,1   3,3,3,1  3,3,2,2  3,3,2,1,1
  3,3,1,1,1,1   3,2,2,2,1  3,2,2,1,1,1   3,2,1,1,1,1,1   3,1,1,1,1,1,1,1   2,2,2,2,2   2,2,2,2,1,1
  2,2,2,1,1,1,1  2,2,1,1,1,1,1,1  2,1,1,1,1,1,1,1,1  1,1,1,1,1,1,1,1,1,1.
-----------------------------------------------------------------------------------------------------------
(End)

Cf. A003107, A000045, A239512.

f = Table[Fibonacci[n], {n, 2, 60}]; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; s[n_, k_] := If[Union[f, DeleteDuplicates[p[n, k]]] == f, p[n, k], 0]; t[n_] := Table[s[n, k], {k, 1, PartitionsP[n]}]; TableForm[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* shows partitions *)
y = Flatten[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* A239001 *)
(* also *)
FibonacciQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]]; Attributes[FibonacciQ] = {Listable}; TableForm[t = Map[Select[IntegerPartitions[#], And @@ FibonacciQ[#] &] &, Range[0, 12]]]
Flatten[t] (* Peter J. C. Moses, Mar 24 2014 *)

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A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

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