A325336 - OEIS

%I #8 Jan 19 2023 02:04:24

%S 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,

%T 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,

%U 1,0,8,3,0,0,0,0,0,0,0,0,1,2,6,6,0,0,0

%N 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.

%C 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).

%H Andrew Howroyd, <a href="/A325336/b325336.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%e Triangle begins:

%e   1

%e   0  1

%e   0  0  1

%e   0  0  1  1

%e   0  0  1  0  1

%e   0  0  1  0  2  0

%e   0  0  1  2  1  0  0

%e   0  0  1  0  3  1  0  0

%e   0  0  1  0  3  2  0  0  0

%e   0  0  1  1  3  3  0  0  0  0

%e   0  0  1  1  5  3  0  0  0  0  0

%e   0  0  1  0  8  3  0  0  0  0  0  0

%e   0  0  1  2  6  6  0  0  0  0  0  0  0

%e   0  0  1  0 13  4  0  0  0  0  0  0  0  0

%e   0  0  1  0 12  8  1  0  0  0  0  0  0  0  0

%e   0  0  1  2 14  7  3  0  0  0  0  0  0  0  0  0

%e   0  0  1  0 17 11  3  0  0  0  0  0  0  0  0  0  0

%e   0  0  1  0 22  7  8  0  0  0  0  0  0  0  0  0  0  0

%e   0  0  1  2 17 16 10  0  0  0  0  0  0  0  0  0  0  0  0

%e   0  0  1  0 28 10 15  0  0  0  0  0  0  0  0  0  0  0  0  0

%e   0  0  1  1 29 13 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0

%e Row 15 counts the following partitions:

%e   111111111111111  54321       433221          333321        4322211

%e                    2222211111  443211          3332211       4332111

%e                                3322221         33222111      43221111

%e                                22222221        322221111

%e                                32222211        332211111

%e                                33321111        432111111

%e                                222222111       321111111111

%e                                3222111111

%e                                3321111111

%e                                22221111111

%e                                32211111111

%e                                222111111111

%e                                2211111111111

%e                                21111111111111

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];

%t Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==k&]],{n,0,30},{k,0,n}]

%o (PARI)

%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 isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}

%o row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}

%o { for(n=0, 10, print(row(n))) } \\ _Andrew Howroyd_, Jan 18 2023

%Y Row sums are A000009.

%Y Column k = 3 is A325334.

%Y Column k = 4 is A325335.

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

%K nonn,tabl

%O 0,20

%A _Gus Wiseman_, May 01 2019