OFFSET
0,13
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. 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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
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 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 1 44 9 1 0 0 0 0 0 0 1
0 1 57 14 3 0 1 0 0 0 0 0 1
0 1 76 20 3 0 0 0 0 0 0 0 0 1
Row n = 8 counts the following partitions.
(8) (44) (431) (2222) (11111111)
(53) (521)
(62)
(71)
(332)
(422)
(611)
(3221)
(3311)
(4211)
(5111)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Switch[#, {}, 0, {_}, 1, _, NestWhile[Sort[Length/@Split[#]]&, #, Length[#]>1&]//First]==k&]], {n, 0, 10}, {k, 0, n}]
PROG
(PARI)
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)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1 + omicron(Vec(p))]++); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023
CROSSREFS
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).
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 18 2019
STATUS
approved