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A325268
Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.
25
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
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
Row sums are A000041. Column k = 2 is A325267.
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).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
Sequence in context: A183700 A275478 A248678 * A232630 A331569 A341716
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 18 2019
STATUS
approved