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A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining. 6
1, 1, 2, 5, 16, 54, 212, 834, 3558, 15394, 69512, 313107, 1474095, 6877031 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of integer partitions finer than (n, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + n ordered by refinement.

a(n+1)/a(n) appears to converge as n -> oo. - Chai Wah Wu, Nov 14 2018

LINKS

Table of n, a(n) for n=0..13.

EXAMPLE

The a(1) = 1 through a(4) = 16 partitions:

  (1)  (21)   (321)     (4321)

       (111)  (2211)    (32221)

              (3111)    (33211)

              (21111)   (42211)

              (111111)  (43111)

                        (222211)

                        (322111)

                        (331111)

                        (421111)

                        (2221111)

                        (3211111)

                        (4111111)

                        (22111111)

                        (31111111)

                        (211111111)

                        (1111111111)

The partition (222211) is the combination of (22)(21)(2)(1), so is counted under a(4). The partition (322111) is the combination of (22)(3)(11)(1), (31)(21)(2)(1), or (211)(3)(2)(1), so is also counted under a(4).

MATHEMATICA

Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1, n]]]], {n, 6}]

CROSSREFS

Cf. A000217, A001970, A002846, A063834, A066723, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.

Cf. A321467, A321468, A321471, A321472, A321514.

Sequence in context: A148400 A052836 A108526 * A149968 A149969 A051960

Adjacent sequences:  A321467 A321468 A321469 * A321471 A321472 A321473

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 11 2018

EXTENSIONS

a(9)-a(11) from Alois P. Heinz, Nov 12 2018

a(12)-a(13) from Chai Wah Wu, Nov 13 2018

STATUS

approved

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Last modified June 14 01:29 EDT 2021. Contains 345016 sequences. (Running on oeis4.)