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A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

%I A336142 #19 Feb 13 2024 20:26:53
%S A336142 1,1,1,4,6,11,22,41,72,142,260,454,769,1416,2472,4465,7708,13314,
%T A336142 23630,40406,68196,119646,203237,343242,586508,993764,1677187,2824072,
%U A336142 4753066,7934268,13355658,22229194,36945828,61555136,102019156,168474033,279181966
%N A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.
%C A336142 A strict composition of n is a finite sequence of distinct positive integers summing to n.
%H A336142 Alois P. Heinz, <a href="/A336142/b336142.txt">Table of n, a(n) for n = 0..7725</a>
%F A336142 G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).
%e A336142 The a(1) = 1 through a(5) = 11 ways:
%e A336142   (1)  (2)  (3)      (4)        (5)
%e A336142             (1,2)    (1,3)      (1,4)
%e A336142             (2,1)    (3,1)      (2,3)
%e A336142             (2),(1)  (3),(1)    (3,2)
%e A336142                      (1,2),(1)  (4,1)
%e A336142                      (2,1),(1)  (3),(2)
%e A336142                                 (4),(1)
%e A336142                                 (1,2),(2)
%e A336142                                 (1,3),(1)
%e A336142                                 (2,1),(2)
%e A336142                                 (3,1),(1)
%p A336142 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A336142      `if`(n=0, p!, b(n, i-1, p)+b(n-i, min(n-i, i-1), p+1)))
%p A336142     end:
%p A336142 g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A336142      `if`(n=0, 1, g(n, i-1)+b(i$2, 0)*g(n-i, min(n-i, i-1))))
%p A336142     end:
%p A336142 a:= n-> g(n$2):
%p A336142 seq(a(n), n=0..38);  # _Alois P. Heinz_, Jul 31 2020
%t A336142 strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
%t A336142 Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
%t A336142 (* Second program: *)
%t A336142 b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
%t A336142      If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
%t A336142 g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
%t A336142      If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
%t A336142 a[n_] := g[n, n];
%t A336142 a /@ Range[0, 38] (* _Jean-François Alcover_, May 20 2021, after _Alois P. Heinz_ *)
%Y A336142 Multiset partitions of partitions are A001970.
%Y A336142 Strict compositions are counted by A032020, A072574, and A072575.
%Y A336142 Splittings of partitions are A323583.
%Y A336142 Splittings of partitions with distinct sums are A336131.
%Y A336142 Cf. A008289, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
%Y A336142 Partitions:
%Y A336142 - Partitions of each part of a partition are A063834.
%Y A336142 - Compositions of each part of a partition are A075900.
%Y A336142 - Strict partitions of each part of a partition are A270995.
%Y A336142 - Strict compositions of each part of a partition are A336141.
%Y A336142 Strict partitions:
%Y A336142 - Partitions of each part of a strict partition are A271619.
%Y A336142 - Compositions of each part of a strict partition are A304961.
%Y A336142 - Strict partitions of each part of a strict partition are A279785.
%Y A336142 - Strict compositions of each part of a strict partition are A336142.
%Y A336142 Compositions:
%Y A336142 - Partitions of each part of a composition are A055887.
%Y A336142 - Compositions of each part of a composition are A133494.
%Y A336142 - Strict partitions of each part of a composition are A304969.
%Y A336142 - Strict compositions of each part of a composition are A307068.
%Y A336142 Strict compositions:
%Y A336142 - Partitions of each part of a strict composition are A336342.
%Y A336142 - Compositions of each part of a strict composition are A336127.
%Y A336142 - Strict partitions of each part of a strict composition are A336343.
%Y A336142 - Strict compositions of each part of a strict composition are A336139.
%K A336142 nonn
%O A336142 0,4
%A A336142 _Gus Wiseman_, Jul 18 2020