A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.

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

Offset

0,13

Comments

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).

Links

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

Example

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)

Mathematica

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}]

Prog

(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

Crossrefs

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

Keyword

nonn,tabl

Author

Gus Wiseman, Apr 18 2019

Status

approved