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A325414
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.
5
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).
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).
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
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Apr 24 2019
STATUS
approved