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A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k. 35
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 (list; table; graph; refs; listen; history; text; internal format)
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 (A325280).

LINKS

Table of n, a(n) for n=0..83.

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}]

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 A325280 (length/frequency depth).

Sequence in context: A025897 A029421 A156749 * A039803 A147809 A325144

Adjacent sequences:  A325277 A325278 A325279 * A325281 A325282 A325283

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Apr 18 2019

STATUS

approved

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Last modified December 11 07:47 EST 2019. Contains 329914 sequences. (Running on oeis4.)