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A325268
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Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.
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25
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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
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OFFSET
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0,13
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COMMENTS
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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.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Switch[#, {}, 0, {_}, 1, _, NestWhile[Sort[Length/@Split[#]]&, #, Length[#]>1&]//First]==k&]], {n, 0, 10}, {k, 0, n}]
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PROG
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(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
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CROSSREFS
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Row sums are A000041. Column k = 2 is A325267.
Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
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
Adjacent sequences: A325265 A325266 A325267 * A325269 A325270 A325271
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KEYWORD
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nonn,tabl
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AUTHOR
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Gus Wiseman, Apr 18 2019
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STATUS
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approved
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