OFFSET
0,20
COMMENTS
The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 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).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
1
0 1
0 0 1
0 0 1 1
0 0 1 0 1
0 0 1 0 2 0
0 0 1 2 1 0 0
0 0 1 0 3 1 0 0
0 0 1 0 3 2 0 0 0
0 0 1 1 3 3 0 0 0 0
0 0 1 1 5 3 0 0 0 0 0
0 0 1 0 8 3 0 0 0 0 0 0
0 0 1 2 6 6 0 0 0 0 0 0 0
0 0 1 0 13 4 0 0 0 0 0 0 0 0
0 0 1 0 12 8 1 0 0 0 0 0 0 0 0
0 0 1 2 14 7 3 0 0 0 0 0 0 0 0 0
0 0 1 0 17 11 3 0 0 0 0 0 0 0 0 0 0
0 0 1 0 22 7 8 0 0 0 0 0 0 0 0 0 0 0
0 0 1 2 17 16 10 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 28 10 15 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 29 13 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Row 15 counts the following partitions:
111111111111111 54321 433221 333321 4322211
2222211111 443211 3332211 4332111
3322221 33222111 43221111
22222221 322221111
32222211 332211111
33321111 432111111
222222111 321111111111
3222111111
3321111111
22221111111
32211111111
222111111111
2211111111111
21111111111111
MATHEMATICA
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#1]]&, ptn, Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n], normQ[#]&&fdadj[#]==k&]], {n, 0, 30}, {k, 0, n}]
PROG
(PARI)
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)}
isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}
row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 01 2019
STATUS
approved