cp's OEIS Frontend

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.

%I A321470 #26 Sep 20 2023 19:11:14
%S A321470 1,1,2,5,16,54,212,834,3558,15394,69512,313107,1474095,6877031,
%T A321470 32877196
%N 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.
%C A321470 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 A321470 a(n+1)/a(n) appears to converge as n -> oo. - _Chai Wah Wu_, Nov 14 2018
%F A321470 a(n) <= A173519(n). - _David A. Corneth_, Sep 20 2023
%e A321470 The a(1) = 1 through a(4) = 16 partitions:
%e A321470   (1)  (21)   (321)     (4321)
%e A321470        (111)  (2211)    (32221)
%e A321470               (3111)    (33211)
%e A321470               (21111)   (42211)
%e A321470               (111111)  (43111)
%e A321470                         (222211)
%e A321470                         (322111)
%e A321470                         (331111)
%e A321470                         (421111)
%e A321470                         (2221111)
%e A321470                         (3211111)
%e A321470                         (4111111)
%e A321470                         (22111111)
%e A321470                         (31111111)
%e A321470                         (211111111)
%e A321470                         (1111111111)
%e A321470 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 A321470 Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1,n]]]],{n,6}]
%o A321470 (Python)
%o A321470 from collections import Counter
%o A321470 from itertools import count, islice
%o A321470 from sympy.utilities.iterables import partitions
%o A321470 def A321470_gen(): # generator of terms
%o A321470     aset = {(1,)}
%o A321470     yield 1
%o A321470     for n in count(2):
%o A321470         yield len(aset)
%o A321470         aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}
%o A321470 A321470_list = list(islice(A321470_gen(),10)) # _Chai Wah Wu_, Sep 20 2023
%Y A321470 Cf. A000217, A001970, A002846, A063834, A066723, A173519, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.
%Y A321470 Cf. A321467, A321468, A321471, A321472, A321514.
%K A321470 nonn,more
%O A321470 0,3
%A A321470 _Gus Wiseman_, Nov 11 2018
%E A321470 a(9)-a(11) from _Alois P. Heinz_, Nov 12 2018
%E A321470 a(12)-a(13) from _Chai Wah Wu_, Nov 13 2018
%E A321470 a(14) from _Chai Wah Wu_, Sep 20 2023