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Number of finite sequences of distinct integer compositions with total sum n.
8

%I #11 Dec 15 2022 17:43:29

%S 1,1,2,8,18,54,156,412,1168,3200,8848,24192,66632,181912,495536,

%T 1354880,3680352,9997056,27093216,73376512,198355840,535319168,

%U 1443042688,3884515008,10445579840,28046885824,75225974912,201536064896,539339293824,1441781213952

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

%H Alois P. Heinz, <a href="/A358907/b358907.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(1) = 1 through a(4) = 18 sequences:

%e ((1)) ((2)) ((3)) ((4))

%e ((11)) ((12)) ((13))

%e ((21)) ((22))

%e ((111)) ((31))

%e ((1)(2)) ((112))

%e ((2)(1)) ((121))

%e ((1)(11)) ((211))

%e ((11)(1)) ((1111))

%e ((1)(3))

%e ((3)(1))

%e ((1)(12))

%e ((11)(2))

%e ((1)(21))

%e ((12)(1))

%e ((2)(11))

%e ((21)(1))

%e ((1)(111))

%e ((111)(1))

%p g:= proc(n) option remember; ceil(2^(n-1)) end:

%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->

%p add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..32); # _Alois P. Heinz_, Dec 15 2022

%t comps[n_]:=Join@@Permutations/@IntegerPartitions[n];

%t Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]

%Y For sets instead of sequences we have A098407, partitions A261049.

%Y This is the strict case of A133494.

%Y The case of distinct sums is A336127, constant sums A074854.

%Y The version for sequences of partitions is A358906.

%Y A001970 counts multiset partitions of integer partitions.

%Y A063834 counts twice-partitions.

%Y A218482 counts sequences of compositions with weakly decreasing lengths.

%Y A358830 counts twice-partitions with distinct lengths.

%Y A358901 counts partitions with all different Omegas.

%Y A358914 counts twice-partitions into distinct strict partitions.

%Y Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 07 2022

%E a(16)-a(29) from _Alois P. Heinz_, Dec 15 2022