Search: a369927 -id:a369927
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A307316
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Number of unlabeled leafless loopless multigraphs with n edges.
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+10
6
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1, 0, 1, 2, 5, 11, 34, 87, 279, 897, 3129, 11458, 44576, 181071, 770237, 3407332, 15641159, 74270464, 364014060, 1837689540, 9540175803, 50853577811, 277976050975, 1556372791835, 8916484189284, 52220798342832, 312389223102731, 1907282708797831, 11876576923779692, 75376983176576501, 487295169002095058
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OFFSET
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0,4
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COMMENTS
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Multigraphs with no loops and no vertices of degree 1.
The initial terms were computed with Nauty.
Conjecturally, the asymptotic number of completely symmetric polynomials of degree n up to momentum conservation in the limit as the number of particles increases.
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LINKS
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FORMULA
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EXAMPLE
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For n=4 the multigraphs (as sets of edges) are {(0,1),(1,2),(2,3),(3,0)}, {(0,1),(0,1),(1,2),(2,0)}, {(0,1),(0,1),(0,1),(0,1)}, {(0,1),(0,1),(1,2),(1,2)}, and {(0,1),(0,1),(2,3),(2,3)}.
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PROG
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(PARI) \\ See also A370063 for a more efficient program.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
seq(n)={my(s=0); forpart(p=2*n, s+=permcount(p)*prod(i=1, #p, 1-x^p[i])/edges(p, w->1-x^w + O(x*x^n))); Vec(s/(2*n)!)} \\ Andrew Howroyd, Feb 01 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A369926
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Number of non-isomorphic set multipartitions (multisets of sets) of weight n without endpoints or singletons.
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+10
2
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1, 0, 0, 0, 1, 0, 3, 1, 9, 8, 34, 45, 177, 324, 1048, 2566, 8050, 22840, 73562, 231978, 780221, 2653042, 9377141, 33820014, 125473936, 475719042, 1846424607, 7317819857, 29611827086, 122190972442, 513900819816, 2201109101784, 9595815668795, 42553843201446, 191861748624324, 879049648551947
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refs;
listen;
history;
text;
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OFFSET
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0,7
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COMMENTS
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A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.
a(n) is also the number of binary matrices with a total of n 1's and every row and column sum at least 2 up to permutation of rows and columns.
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LINKS
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EXAMPLE
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The a(8) = 9 matrices are:
[1 1 1 1] [1 1 1] [1 1 1 0] [1 1 1 1]
[1 1 1 1] [1 1 1] [1 1 0 1] [1 1 0 0]
[1 1 0] [0 0 1 1] [0 0 1 1]
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[1 1] [1 1 0] [1 1 0] [1 1 0 0] [1 1 0 0]
[1 1] [1 1 0] [1 1 0] [1 1 0 0] [1 0 1 0]
[1 1] [1 0 1] [1 0 1] [0 0 1 1] [0 1 0 1]
[1 1] [1 0 1] [0 1 1] [0 0 1 1] [0 0 1 1]
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PROG
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(PARI) Vec(G(25, 1)) \\ G defined in A369927.
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CROSSREFS
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A321677 is the case without singletons but allowing endpoints (or by duality without endpoints but allowing singletons).
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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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