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