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%I #7 Apr 25 2019 13:30:51
%S 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,
%T 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,
%U 3,1,0,1,0,0,1,4,0,3,3,3,4,1,6,3,1,0,1
%N Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.
%C 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).
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 0 1
%e 0 1 0 0 1 1
%e 0 1 0 1 0 2 0 0 1
%e 0 1 0 0 0 2 1 0 2 1
%e 0 1 0 1 1 2 0 3 1 1 1
%e 0 1 0 0 0 3 0 1 4 2 2 1 1
%e 0 1 0 1 0 4 0 3 3 2 2 2 3 1
%e 0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
%e 0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
%e Row n = 9 counts the following partitions:
%e 9 333 54 432 441 3222 22221 411111 3321 32211 321111
%e 63 531 522 6111 33111 4221 42111
%e 72 621 711 222111 51111 4311 21111111
%e 81 111111111 5211
%e 2211111
%e 3111111
%t omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t Table[Length[Select[IntegerPartitions[n],Total[omseq[#]]==k&]],{n,0,10},{k,0,Max[Total/@omseq/@IntegerPartitions[n]]}]
%Y Row sums are A000041.
%Y Row lengths are A325413(n) + 1 (because k starts at 0).
%Y Number of nonzero terms in row n is A325415(n).
%Y Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.
%Y 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).
%K nonn,tabf
%O 0,19
%A _Gus Wiseman_, Apr 24 2019
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