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A348532 a(n) is the number of multisets of integers that are possible to reach by starting with n occurrences of 0 and by splitting and reverse splitting.

%I A348532 #81 Nov 19 2021 07:44:42
%S A348532 1,1,2,2,7,9,43,59,338,490,3097,4639,31283,48107,338553,531469,
%T A348532 3857036,6157068,45713546,73996100
%N A348532 a(n) is the number of multisets of integers that are possible to reach by starting with n occurrences of 0 and by splitting and reverse splitting.
%C A348532 Splitting is taking 2 occurrences of the same integer and incrementing one of them by 1 and decrementing the other occurrence by 1.
%C A348532 Reverse splitting is taking two elements with a difference of 2 and incrementing the smaller one by 1 and decrementing the larger one by 1. It is the opposite of splitting.
%F A348532 It appears that a(n) = A000571(n) for odd n.
%e A348532 For n = 5, the multisets are as follows:
%e A348532   {{0,0,0,0,0}}   {{-1,0,0,0,1}}   {{-1,-1,0,1,1}}
%e A348532   {{-1,-1,0,0,2}} {{-1,-1,-1,1,2}} {{-2,0,0,1,1}}
%e A348532   {{-2,0,0,0,2}}  {{-2,-1,1,1,1}}  {{-2,-1,0,1,2}}.
%e A348532   Therefore, a(5) = 9.
%e A348532 For n = 6, the multisets are as follows:
%e A348532   {{0,0,0,0,0,0}}     {{-1,0,0,0,0,1}}     {{-1,-1,0,0,1,1}}
%e A348532   {{-1,-1,0,0,0,2}}   {{-1,-1,-1,1,1,1}}   {{-1,-1,-1,0,1,2}}
%e A348532   {{-1,-1,-1,0,0,3}}* {{-1,-1,-1,-1,2,2}}* {{-1,-1,-1,-1,1,3}}*
%e A348532   {{-2,0,0,0,1,1}}    {{-2,0,0,0,0,2}}     {{-2,-1,0,1,1,1}}
%e A348532   {{-2,-1,0,0,1,2}}   {{-2,-1,0,0,0,3}}*   {{-2,-1,-1,1,1,2}}
%e A348532   {{-2,-1,-1,0,2,2}}  {{-2,-1,-1,0,1,3}}   {{-2,-1,-1,-1,2,3}}*
%e A348532   {{-2,-2,1,1,1,1}}*  {{-2,-2,0,1,1,2}}    {{-2,-2,0,0,2,2}}
%e A348532   {{-2,-2,0,0,1,3}}   {{-2,-2,-1,1,2,2}}   {{-2,-2,-1,1,1,3}}
%e A348532   {{-2,-2,-1,0,2,3}}  {{-2,-2,-2,2,2,2}}*  {{-2,-2,-2,1,2,3}}*
%e A348532   {{-3,0,0,0,0,3}}*   {{-3,0,0,0,1,2}}*    {{-3,0,0,1,1,1}}*
%e A348532   {{-3,-1,1,1,1,1}}*  {{-3,-1,0,1,1,2}}    {{-3,-1,0,0,2,2}}
%e A348532   {{-3,-1,0,0,1,3}}   {{-3,-1,-1,1,2,2}}   {{-3,-1,-1,1,1,3}}
%e A348532   {{-3,-1,-1,0,2,3}}  {{-3,-2,1,1,1,2}}*   {{-3,-2,0,1,2,2}}
%e A348532   {{-3,-2,0,1,1,3}}   {{-3,-2,0,0,2,3}}    {{-3,-2,-1,2,2,2}}*
%e A348532   {{-3,-2,-1,1,2,3}}.
%e A348532   Therefore, a(6) = 43.
%e A348532 The ones marked with an asterisk are the ones that need reverse splitting
%e A348532 to be reached. They are not produced using the rules of A347913.
%o A348532 (Python)
%o A348532 def nextq(q):
%o A348532     used, used2 = set(), set()
%o A348532     for i in range(len(q)-1):
%o A348532         for j in range(i+1, len(q)):
%o A348532             if q[i] == q[j]:
%o A348532                 if q[i] in used: continue
%o A348532                 used.add(q[i])
%o A348532                 qc = list(q); qc[i] -= 1; qc[j] += 1
%o A348532                 yield tuple(sorted(qc))
%o A348532             elif q[j] - q[i] == 2:  # assumes q is sorted
%o A348532                 if q[i] in used2: continue
%o A348532                 used2.add(q[i])
%o A348532                 qc = list(q); qc[i] += 1; qc[j] -= 1
%o A348532                 yield tuple(sorted(qc))
%o A348532 def a(n):
%o A348532     s = tuple(0 for i in range(n)); reach = {s}; expand = list(reach)
%o A348532     while len(expand) > 0:
%o A348532         q = expand.pop()
%o A348532         for qq in nextq(q):
%o A348532             if qq not in reach:
%o A348532                 reach.add(qq)
%o A348532                 expand.append(qq)
%o A348532     return len(reach)
%o A348532 print([a(n) for n in range(13)]) # _Michael S. Branicky_, Oct 21 2021
%Y A348532 Cf. A000571, A347913.
%K A348532 nonn,more
%O A348532 0,3
%A A348532 _Tejo Vrush_, Oct 21 2021
%E A348532 a(6)-a(19) from _Michael S. Branicky_, Oct 21 2021