cp's OEIS Frontend

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

%I A325414 #7 Apr 25 2019 13:30:51
%S A325414 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 A325414 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 A325414 3,1,0,1,0,0,1,4,0,3,3,3,4,1,6,3,1,0,1
%N A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.
%C A325414 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 A325414 Triangle begins:
%e A325414   1
%e A325414   0 1
%e A325414   0 1 0 1
%e A325414   0 1 0 0 1 1
%e A325414   0 1 0 1 0 2 0 0 1
%e A325414   0 1 0 0 0 2 1 0 2 1
%e A325414   0 1 0 1 1 2 0 3 1 1 1
%e A325414   0 1 0 0 0 3 0 1 4 2 2 1 1
%e A325414   0 1 0 1 0 4 0 3 3 2 2 2 3 1
%e A325414   0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
%e A325414   0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
%e A325414 Row n = 9 counts the following partitions:
%e A325414   9  333  54  432  441  3222    22221      411111  3321     32211     321111
%e A325414           63  531  522  6111    33111              4221     42111
%e A325414           72  621  711  222111  51111              4311     21111111
%e A325414           81                    111111111          5211
%e A325414                                                    2211111
%e A325414                                                    3111111
%t A325414 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325414 Table[Length[Select[IntegerPartitions[n],Total[omseq[#]]==k&]],{n,0,10},{k,0,Max[Total/@omseq/@IntegerPartitions[n]]}]
%Y A325414 Row sums are A000041.
%Y A325414 Row lengths are A325413(n) + 1 (because k starts at 0).
%Y A325414 Number of nonzero terms in row n is A325415(n).
%Y A325414 Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.
%Y A325414 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 A325414 nonn,tabf
%O A325414 0,19
%A A325414 _Gus Wiseman_, Apr 24 2019