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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..86.

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}]

CROSSREFS

Row sums are A000009.

Column k = 3 is A325334.

Column k = 4 is A325335.

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

Sequence in context: A089605 A218786 A218787 * A060016 A117408 A228360

Adjacent sequences:  A325333 A325334 A325335 * A325337 A325338 A325339

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, May 01 2019

STATUS

approved

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Last modified November 14 20:15 EST 2019. Contains 329130 sequences. (Running on oeis4.)