cp's OEIS Frontend

A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

%I A374704 #9 Dec 29 2024 18:18:39
%S A374704 1,1,3,6,15,31,77,171,410,957,2275,5370,12795,30366,72307,172071,
%T A374704 409875,976155,2325804,5541230,13204161,31464226,74980838,178684715,
%U A374704 425830008,1014816979,2418489344,5763712776,13736075563,32735874251,78016456122,185929792353,443110675075
%N A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.
%H A374704 Andrew Howroyd, <a href="/A374704/b374704.txt">Table of n, a(n) for n = 0..1000</a>
%F A374704 G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - _Andrew Howroyd_, Dec 29 2024
%e A374704 The a(0) = 1 through a(4) = 15 ways:
%e A374704   ()  ((1))  ((2))      ((3))          ((4))
%e A374704              ((1,1))    ((1,2))        ((1,3))
%e A374704              ((1),(1))  ((1,1,1))      ((2,2))
%e A374704                         ((1),(1,1))    ((1,1,2))
%e A374704                         ((1,1),(1))    ((2),(2))
%e A374704                         ((1),(1),(1))  ((1,1,1,1))
%e A374704                                        ((1),(1,2))
%e A374704                                        ((1,2),(1))
%e A374704                                        ((1),(1,1,1))
%e A374704                                        ((1,1),(1,1))
%e A374704                                        ((1,1,1),(1))
%e A374704                                        ((1),(1),(1,1))
%e A374704                                        ((1),(1,1),(1))
%e A374704                                        ((1,1),(1),(1))
%e A374704                                        ((1),(1),(1),(1))
%t A374704 Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]
%o A374704 (PARI) seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ _Andrew Howroyd_, Dec 29 2024
%Y A374704 A variation for weakly increasing lengths is A141199.
%Y A374704 For identical sums instead of minima we have A279787.
%Y A374704 The case of reversed twice-partitions is A306319, distinct A358830.
%Y A374704 For maxima instead of minima, or for unreversed partitions, we have A358905.
%Y A374704 The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).
%Y A374704 A003242 counts anti-run compositions, ranks A333489.
%Y A374704 A011782 counts compositions.
%Y A374704 A238130, A238279, A333755 count compositions by number of runs.
%Y A374704 A274174 counts contiguous compositions, ranks A374249.
%Y A374704 A055887 counts sequences of partitions with total sum n.
%Y A374704 A281145 counts same-trees.
%Y A374704 A319169 counts partitions with constant Omega, ranked by A320324.
%Y A374704 A358911 counts compositions with constant Omega, distinct A358912.
%Y A374704 Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.
%K A374704 nonn
%O A374704 0,3
%A A374704 _Gus Wiseman_, Aug 04 2024
%E A374704 a(16) onwards from _Andrew Howroyd_, Dec 29 2024