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

1,2

Comments

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

Links

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

Example

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

Mathematica

nrows=10; Array[ReverseSort[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), , 4); \\ Michel Marcus, Jun 16 2022

Crossrefs

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.

Sequence in context: A330370 A330371 A080577 * A209655 A209918 A030312

Adjacent sequences: A302243 A302244 A302245 * A302247 A302248 A302249

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 05 2018

Status

approved