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A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

%I A325268 #12 Jan 19 2023 11:10:24
%S A325268 1,0,1,0,1,1,0,1,1,1,0,1,3,0,1,0,1,5,0,0,1,0,1,7,2,0,0,1,0,1,12,1,0,0,
%T A325268 0,1,0,1,17,2,1,0,0,0,1,0,1,24,4,0,0,0,0,0,1,0,1,33,5,1,1,0,0,0,0,1,0,
%U A325268 1,44,9,1,0,0,0,0,0,0,1,0,1,57,14,3,0,1
%N A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.
%C A325268 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. The omicron of the partition is 0 if the omega-sequence is empty, 1 if it is a singleton, and otherwise the second-to-last part. 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), and its omicron is 2.
%H A325268 Andrew Howroyd, <a href="/A325268/b325268.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A325268 Triangle begins:
%e A325268   1
%e A325268   0  1
%e A325268   0  1  1
%e A325268   0  1  1  1
%e A325268   0  1  3  0  1
%e A325268   0  1  5  0  0  1
%e A325268   0  1  7  2  0  0  1
%e A325268   0  1 12  1  0  0  0  1
%e A325268   0  1 17  2  1  0  0  0  1
%e A325268   0  1 24  4  0  0  0  0  0  1
%e A325268   0  1 33  5  1  1  0  0  0  0  1
%e A325268   0  1 44  9  1  0  0  0  0  0  0  1
%e A325268   0  1 57 14  3  0  1  0  0  0  0  0  1
%e A325268   0  1 76 20  3  0  0  0  0  0  0  0  0  1
%e A325268 Row n = 8 counts the following partitions.
%e A325268   (8)  (44)       (431)  (2222)  (11111111)
%e A325268        (53)       (521)
%e A325268        (62)
%e A325268        (71)
%e A325268        (332)
%e A325268        (422)
%e A325268        (611)
%e A325268        (3221)
%e A325268        (3311)
%e A325268        (4211)
%e A325268        (5111)
%e A325268        (22211)
%e A325268        (32111)
%e A325268        (41111)
%e A325268        (221111)
%e A325268        (311111)
%e A325268        (2111111)
%t A325268 Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{_},1,_,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==k&]],{n,0,10},{k,0,n}]
%o A325268 (PARI)
%o A325268 omicron(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); r=#p; for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L); r)}
%o A325268 row(n)={my(v=vector(1+n)); forpart(p=n, v[1 + omicron(Vec(p))]++); v}
%o A325268 { for(n=0, 10, print(row(n))) } \\ _Andrew Howroyd_, Jan 18 2023
%Y A325268 Row sums are A000041. Column k = 2 is A325267.
%Y A325268 Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
%Y A325268 Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
%Y A325268 Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K A325268 nonn,tabl
%O A325268 0,13
%A A325268 _Gus Wiseman_, Apr 18 2019