A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

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

Offset: 0

Author

Gus Wiseman, May 01 2019

Keywords

• nonn
• tabl

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).

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Row sums are A000009.

Column k = 3 is A325334.

Column k = 4 is A325335.

Cf. A181819, A182850, A317246, A320348, A323014, A325280, A325372.

Programs

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}]

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