A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

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

Offset

0,19

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13, so (32211) is counted under T(9,13).

Links

Table of n, a(n) for n=0..86.

Example

Triangle begins:
  1
  0 1
  0 1 0 1
  0 1 0 0 1 1
  0 1 0 1 0 2 0 0 1
  0 1 0 0 0 2 1 0 2 1
  0 1 0 1 1 2 0 3 1 1 1
  0 1 0 0 0 3 0 1 4 2 2 1 1
  0 1 0 1 0 4 0 3 3 2 2 2 3 1
  0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
  0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
Row n = 9 counts the following partitions:
  9  333  54  432  441  3222    22221      411111  3321     32211     321111
          63  531  522  6111    33111              4221     42111
          72  621  711  222111  51111              4311     21111111
          81                    111111111          5211
                                                   2211111
                                                   3111111

Mathematica

omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n], Total[omseq[#]]==k&]], {n, 0, 10}, {k, 0, Max[Total/@omseq/@IntegerPartitions[n]]}]

Crossrefs

Row sums are A000041.

Row lengths are A325413(n) + 1 (because k starts at 0).

Number of nonzero terms in row n is A325415(n).

Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Sequence in context: A358434 A030199 A320005 * A216510 A005089 A364127

Adjacent sequences: A325411 A325412 A325413 * A325415 A325416 A325417

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 24 2019

Status

approved