Search: a125699 -id:a125699
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A125702
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Number of connected categories with n objects and 2n-1 morphisms.
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+10
6
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1, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793
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listen;
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internal format)
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OFFSET
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1,3
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COMMENTS
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Also number of connected antitransitive relations on n objects (antitransitive meaning a R b and b R c implies not a R c); equivalently, number of free oriented bipartite trees, with all arrows going from one part to the other part.
Also the number of non-isomorphic multi-hypertrees of weight n - 1 with singletons allowed. A multi-hypertree with singletons allowed is a connected set multipartition (multiset of sets) with density -1, where the density of a set multipartition is the weight (sum of sizes of the parts) minus the number of parts minus the number of vertices. - Gus Wiseman, Oct 30 2018
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LINKS
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FORMULA
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EXAMPLE
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Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
{} {{1}} {{12}} {{123}} {{1234}} {{12345}}
{{1}{1}} {{2}{12}} {{13}{23}} {{14}{234}}
{{1}{1}{1}} {{3}{123}} {{4}{1234}}
{{1}{2}{12}} {{2}{13}{23}}
{{2}{2}{12}} {{2}{3}{123}}
{{1}{1}{1}{1}} {{3}{13}{23}}
{{3}{3}{123}}
{{1}{2}{2}{12}}
{{2}{2}{2}{12}}
{{1}{1}{1}{1}{1}}
(End)
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PROG
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(PARI) \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ Andrew Howroyd, Nov 02 2019
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CROSSREFS
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Same as A122086 except for n = 1; see there for formulas. Cf. A125699.
Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A125700
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Number of connected categories with n more morphisms than objects.
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+10
3
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OFFSET
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0,2
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LINKS
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Eric Weisstein's World of Mathematics, Category
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FORMULA
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a(n) = Sum_{k=1..n+1} A125699(n+k,k).
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A125697
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Table, T(n,k) is the number of categories with n morphisms and k objects.
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+10
2
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1, 2, 1, 7, 3, 1, 35, 16, 3, 1, 228, 77, 20, 3, 1, 2237, 485, 111, 21, 3, 1, 31559, 4013, 716, 127, 21, 3, 1, 1668997, 47648, 5623, 862, 131, 21, 3, 1
(list;
table;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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This is a two-dimensional Euler transform of A125699.
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LINKS
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Eric Weisstein's World of Mathematics, Category
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FORMULA
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G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).
T(n,k) = A125701(n-k) when k >= (2/3)*n.
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.
(End)
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EXAMPLE
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The table starts:
1;
2, 1;
7, 3, 1;
35, 16, 3, 1;
228, 77, 20, 3, 1;
2237, 485, 111, 21, 3, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A125698
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Number of connected categories with n morphisms.
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+10
2
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OFFSET
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1,2
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COMMENTS
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Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.
Inverse Euler transform of A125696.
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LINKS
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Eric Weisstein's World of Mathematics, Category
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EXAMPLE
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The 8 categories with 3 morphisms consist of 7 3-element monoids and one 2-object category with a single morphism between the objects.
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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