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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

%I #26 Sep 20 2023 19:11:14

%S 1,1,2,5,16,54,212,834,3558,15394,69512,313107,1474095,6877031,

%T 32877196

%N 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.

%C 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.

%C a(n+1)/a(n) appears to converge as n -> oo. - _Chai Wah Wu_, Nov 14 2018

%F a(n) <= A173519(n). - _David A. Corneth_, Sep 20 2023

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

%e (1) (21) (321) (4321)

%e (111) (2211) (32221)

%e (3111) (33211)

%e (21111) (42211)

%e (111111) (43111)

%e (222211)

%e (322111)

%e (331111)

%e (421111)

%e (2221111)

%e (3211111)

%e (4111111)

%e (22111111)

%e (31111111)

%e (211111111)

%e (1111111111)

%e 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).

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

%o (Python)

%o from collections import Counter

%o from itertools import count, islice

%o from sympy.utilities.iterables import partitions

%o def A321470_gen(): # generator of terms

%o aset = {(1,)}

%o yield 1

%o for n in count(2):

%o yield len(aset)

%o aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}

%o A321470_list = list(islice(A321470_gen(),10)) # _Chai Wah Wu_, Sep 20 2023

%Y Cf. A000217, A001970, A002846, A063834, A066723, A173519, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Nov 11 2018

%E a(9)-a(11) from _Alois P. Heinz_, Nov 12 2018

%E a(12)-a(13) from _Chai Wah Wu_, Nov 13 2018

%E a(14) from _Chai Wah Wu_, Sep 20 2023