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%I #15 Aug 21 2019 05:32:28
%S 1,1,1,1,2,1,1,7,3,1,1,23,29,4,1,1,122,2102,150,5,1,1,888,7011184,
%T 7013164,1037,6,1,1,11302,1788775603336,29281354507753848,
%U 1788782615612,12338,7,1,1,262322,53304526022885280592,234431745534048893449761040648512,234431745534048922729326772799024,53304527811667884902,274659,8,1
%N Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices.
%H Andrew Howroyd, <a href="/A301922/b301922.txt">Table of n, a(n) for n = 1..91</a>
%F T(n,k) = A309858(n,k) - A309858(n-1,k). - _Alois P. Heinz_, Aug 21 2019
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 7 3 1
%e 1 23 29 4 1
%e The T(4,2) = 7 hypergraphs:
%e {{1,2},{3,4}}
%e {{1,3},{2,4},{3,4}}
%e {{1,4},{2,4},{3,4}}
%e {{1,2},{1,3},{2,4},{3,4}}
%e {{1,4},{2,3},{2,4},{3,4}}
%e {{1,3},{1,4},{2,3},{2,4},{3,4}}
%e {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
%p g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
%p [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
%p h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
%p /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
%p /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
%p `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
%p b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
%p /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
%p A:= proc(n, k) A(n, k):= `if`(k>n-k, A(n, n-k), b(n$2, [], k)) end:
%p T:= (n, k)-> A(n, k)-A(n-1, k):
%p seq(seq(T(n, k), k=1..n), n=1..9); # _Alois P. Heinz_, Aug 21 2019
%o (PARI)
%o 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}
%o rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L,k); while(#L<k, listput(L,#L))); Vec(L)}
%o can(v,f)={my(d=1,u=v); while(d>0, u=vecsort(apply(f, u)); d=lex(u,v)); !d}
%o Q(n,k,perm)={my(t=0); forsubset([n,k], v, t += can(Vec(v), t->perm[t])); t}
%o U(n,k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n,k,rep(p))); s/n!}
%o for(n=1, 10, for(k=1, n, print1(U(n,k)-U(n-1,k), ", ")); print) \\ _Andrew Howroyd_, Aug 10 2019
%Y Row sums are A301481. Second column is A002494.
%Y Cf. A003465, A055621, A298422, A298426, A299471, A301481, A301920, A306017-A306021, A309858.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, Jun 19 2018
%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 09 2019
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