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A325280
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Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.
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50
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 3, 0, 0, 1, 3, 4, 3, 0, 0, 0, 1, 1, 4, 8, 1, 0, 0, 0, 1, 3, 6, 9, 3, 0, 0, 0, 0, 1, 2, 8, 12, 7, 0, 0, 0, 0, 0, 1, 3, 11, 17, 10, 0, 0, 0, 0, 0, 0, 1, 1, 11, 26, 17, 0, 0, 0, 0, 0, 0, 0, 1, 5, 19, 25, 27
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OFFSET
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0,13
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COMMENTS
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The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (this sequence).
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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 2 1 1
0 1 1 2 3 0
0 1 3 4 3 0 0
0 1 1 4 8 1 0 0
0 1 3 6 9 3 0 0 0
0 1 2 8 12 7 0 0 0 0
0 1 3 11 17 10 0 0 0 0 0
0 1 1 11 26 17 0 0 0 0 0 0
0 1 5 19 25 27 0 0 0 0 0 0 0
0 1 1 17 44 38 0 0 0 0 0 0 0 0
0 1 3 25 53 52 1 0 0 0 0 0 0 0 0
0 1 3 29 63 76 4 0 0 0 0 0 0 0 0 0
0 1 4 37 83 98 8 0 0 0 0 0 0 0 0 0 0
Row n = 9 counts the following partitions:
(9) (333) (54) (441) (3321)
(111111111) (63) (522) (4221)
(72) (711) (4311)
(81) (3222) (5211)
(432) (6111) (32211)
(531) (22221) (42111)
(621) (33111) (321111)
(222111) (51111)
(411111)
(2211111)
(3111111)
(21111111)
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MATHEMATICA
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fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n], fdadj[#]==k&]], {n, 0, 16}, {k, 0, n}]
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PROG
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(PARI) \\ depth(p) gives adjusted frequency depth of partition.
depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L, i-k); k=i)); listsort(L); p=L; r++); r)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1+depth(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 A032741. Column k = 3 is A325245.
Cf. A181819, A225486, A323014, A323023, A325254, A325258, A325277.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or this sequence (length/frequency depth).
Sequence in context: A029421 A156749 A340390 * A039803 A360118 A147809
Adjacent sequences: A325277 A325278 A325279 * A325281 A325282 A325283
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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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