cp's OEIS Frontend

A358907 Number of finite sequences of distinct integer compositions with total sum n.

%I A358907 #11 Dec 15 2022 17:43:29
%S A358907 1,1,2,8,18,54,156,412,1168,3200,8848,24192,66632,181912,495536,
%T A358907 1354880,3680352,9997056,27093216,73376512,198355840,535319168,
%U A358907 1443042688,3884515008,10445579840,28046885824,75225974912,201536064896,539339293824,1441781213952
%N A358907 Number of finite sequences of distinct integer compositions with total sum n.
%H A358907 Alois P. Heinz, <a href="/A358907/b358907.txt">Table of n, a(n) for n = 0..1000</a>
%e A358907 The a(1) = 1 through a(4) = 18 sequences:
%e A358907   ((1))  ((2))   ((3))      ((4))
%e A358907          ((11))  ((12))     ((13))
%e A358907                  ((21))     ((22))
%e A358907                  ((111))    ((31))
%e A358907                  ((1)(2))   ((112))
%e A358907                  ((2)(1))   ((121))
%e A358907                  ((1)(11))  ((211))
%e A358907                  ((11)(1))  ((1111))
%e A358907                             ((1)(3))
%e A358907                             ((3)(1))
%e A358907                             ((1)(12))
%e A358907                             ((11)(2))
%e A358907                             ((1)(21))
%e A358907                             ((12)(1))
%e A358907                             ((2)(11))
%e A358907                             ((21)(1))
%e A358907                             ((1)(111))
%e A358907                             ((111)(1))
%p A358907 g:= proc(n) option remember; ceil(2^(n-1)) end:
%p A358907 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
%p A358907       add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
%p A358907     end:
%p A358907 a:= n-> b(n$2, 0):
%p A358907 seq(a(n), n=0..32);  # _Alois P. Heinz_, Dec 15 2022
%t A358907 comps[n_]:=Join@@Permutations/@IntegerPartitions[n];
%t A358907 Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]
%Y A358907 For sets instead of sequences we have A098407, partitions A261049.
%Y A358907 This is the strict case of A133494.
%Y A358907 The case of distinct sums is A336127, constant sums A074854.
%Y A358907 The version for sequences of partitions is A358906.
%Y A358907 A001970 counts multiset partitions of integer partitions.
%Y A358907 A063834 counts twice-partitions.
%Y A358907 A218482 counts sequences of compositions with weakly decreasing lengths.
%Y A358907 A358830 counts twice-partitions with distinct lengths.
%Y A358907 A358901 counts partitions with all different Omegas.
%Y A358907 A358914 counts twice-partitions into distinct strict partitions.
%Y A358907 Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.
%K A358907 nonn
%O A358907 0,3
%A A358907 _Gus Wiseman_, Dec 07 2022
%E A358907 a(16)-a(29) from _Alois P. Heinz_, Dec 15 2022