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A356606 Number of strict integer partitions of n where all parts have neighbors.

%I A356606 #25 Feb 24 2024 10:05:21
%S A356606 1,0,0,1,0,1,1,1,0,2,1,1,2,1,2,3,2,2,5,2,4,5,5,4,8,5,7,9,8,8,13,10,11,
%T A356606 16,13,15,20,18,18,27,21,26,31,30,30,43,34,42,49,48,48,65,56,65,76,74,
%U A356606 77,97,88,98,117,111,119,143,137,146,175,165,182,208
%N A356606 Number of strict integer partitions of n where all parts have neighbors.
%C A356606 A part x has a neighbor if either x - 1 or x + 1 is a part.
%H A356606 Alois P. Heinz, <a href="/A356606/b356606.txt">Table of n, a(n) for n = 0..5000</a> (first 301 terms from John Tyler Rascoe)
%H A356606 John Tyler Rascoe, <a href="/A356606/a356606.py.txt">Python program</a>
%F A356606 G.f.: 1 + Sum_{i>0} A(x,i), where A(x,i) = x^((2*i)+1) * G(x,i+1) for i > 0, is the g.f. for partitions of this kind with least part i, and G(x,k) = 1 + x^(k+1) * G(x,k+1) + Sum_{m>=0} x^(2*(k+m)+5) * G(x,m+k+3). - _John Tyler Rascoe_, Feb 16 2024
%e A356606 The a(n) partitions for n = 0, 1, 3, 9, 15, 18, 20, 24 (A = 10, B = 11):
%e A356606   ()  .  (21)  (54)   (87)     (765)    (7643)   (987)
%e A356606                (432)  (654)    (6543)   (8732)   (8754)
%e A356606                       (54321)  (7632)   (9821)   (9843)
%e A356606                                (8721)   (65432)  (A932)
%e A356606                                (65421)           (BA21)
%e A356606                                                  (87432)
%e A356606                                                  (87621)
%e A356606                                                  (765321)
%t A356606 Table[Length[Select[IntegerPartitions[n], Function[ptn,UnsameQ@@ptn&&And@@Table[MemberQ[ptn,x-1]||MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%o A356606 (Python) # see linked program
%Y A356606 This is the strict case of A355393 and A355394.
%Y A356606 The complement is counted by A356607, non-strict A356235 and A356236.
%Y A356606 A000041 counts integer partitions, strict A000009.
%Y A356606 A000837 counts relatively prime partitions, ranked by A289509.
%Y A356606 A007690 counts partitions with no singletons, complement A183558.
%Y A356606 Cf. A137921, A325160, A328171, A328172, A328187, A328220, A328221, A356237.
%K A356606 nonn
%O A356606 0,10
%A A356606 _Gus Wiseman_, Aug 24 2022