A302247 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nondecreasing order.

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

Offset

1,4

Comments

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

Links

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

Example

Triangle begins:
  1;
  1,1,2;
  1,1,1,1,2,3;
  1,1,1,1,1,1,1,2,2,2,3,4;
  1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There are seven 1's, three 2's, only one 3 and only one 4, so the 4th row of this triangle is [1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4].
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 nondecreasing order give the 4th row of this triangle. - Omar E. Pol, Jun 16 2022

Mathematica

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

Prog

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

Crossrefs

Mirror of A302246.

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 both A026791 and A080576 at a(17) = T(4,7).

Cf. A000041, A027750, A176206, A221529, A336812, A338156.

Sequence in context: A104758 A143227 A329746 * A026791 A080576 A321744

Adjacent sequences: A302244 A302245 A302246 * A302248 A302249 A302250

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 05 2018

Status

approved