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%I #11 Jan 19 2023 22:35:12
%S 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,
%T 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,
%U 0,1,1,11,26,17,0,0,0,0,0,0,0,1,5,19,25,27
%N Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.
%C 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).
%C 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).
%H Andrew Howroyd, <a href="/A325280/b325280.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 2 1 1
%e 0 1 1 2 3 0
%e 0 1 3 4 3 0 0
%e 0 1 1 4 8 1 0 0
%e 0 1 3 6 9 3 0 0 0
%e 0 1 2 8 12 7 0 0 0 0
%e 0 1 3 11 17 10 0 0 0 0 0
%e 0 1 1 11 26 17 0 0 0 0 0 0
%e 0 1 5 19 25 27 0 0 0 0 0 0 0
%e 0 1 1 17 44 38 0 0 0 0 0 0 0 0
%e 0 1 3 25 53 52 1 0 0 0 0 0 0 0 0
%e 0 1 3 29 63 76 4 0 0 0 0 0 0 0 0 0
%e 0 1 4 37 83 98 8 0 0 0 0 0 0 0 0 0 0
%e Row n = 9 counts the following partitions:
%e (9) (333) (54) (441) (3321)
%e (111111111) (63) (522) (4221)
%e (72) (711) (4311)
%e (81) (3222) (5211)
%e (432) (6111) (32211)
%e (531) (22221) (42111)
%e (621) (33111) (321111)
%e (222111) (51111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
%t Table[Length[Select[IntegerPartitions[n],fdadj[#]==k&]],{n,0,16},{k,0,n}]
%o (PARI) \\ depth(p) gives adjusted frequency depth of partition.
%o 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)}
%o row(n)={my(v=vector(1+n)); forpart(p=n, v[1+depth(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 A032741. Column k = 3 is A325245.
%Y Cf. A181819, A225486, A323014, A323023, A325254, A325258, A325277.
%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or this sequence (length/frequency depth).
%K nonn,tabl
%O 0,13
%A _Gus Wiseman_, Apr 18 2019
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