cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 51-53 of 53 results.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190
Offset: 0

Views

Author

Gus Wiseman, Jan 06 2024

Keywords

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

Links

Crossrefs

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Formula

Inverse Euler transform of A339888.

A054922 Number of connected unlabeled symmetric relations (graphs with loops) having n nodes such that complement is also connected.

Original entry on oeis.org

2, 0, 0, 10, 164, 2670, 56724, 1867860, 104538928, 10461483366, 1912179618740, 644464839239880, 402785011941549964, 468944407349226545614, 1021179521951204217530900, 4174755063830188009750183026
Offset: 1

Views

Author

N. J. A. Sloane, May 24 2000

Keywords

Links

Crossrefs

Cf. A000666, A054291.

Programs

  • Mathematica
    A000666 = Cases[Import["https://oeis.org/A000666/b000666.txt", "Table"], {, }][[All, 2]];
A054921 = Cases[Import["https://oeis.org/A054921/b054921.txt", "Table"], {, }][[All, 2]];
a[n_] := 2*A054921[[n + 1]] - A000666[[n + 1]];
Array[a, 50] (* Jean-François Alcover, Aug 31 2019 *)
  • Python
    from functools import lru_cache
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
from sympy import mobius, divisors
def A054922(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return (sum(mobius(d)*c(n//d) for d in divisors(n,generator=True))//n<<1)-b(n) # Chai Wah Wu, Jul 10 2024

Formula

a(n) = 2*A054921(n) - A000666(n).

Extensions

More terms from Vladeta Jovovic, Jul 17 2000

A320993 Number of connected self-dual marked graphs on 2n nodes.

Original entry on oeis.org

1, 1, 6, 81, 2796, 285205, 96322648, 112087066485, 458071927263177, 6665704296474517580, 349377209492189224235030, 66602723163954143548104716149, 46557323273646194397454383970079368, 120168498151800396724425771086539073209571, 1152049915423012273792614840558950392103437052846
Offset: 0

Views

Author

N. J. A. Sloane, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A000666 (not necessarily connected marked graphs), A000595 (self dual on 2n nodes), A054921 (connected marked graphs).

Programs

Formula

a(2*n-1) = b(2*n-1) - A054921(2*n-1)/2, a(2*n) = b(2*n) - (A054921(2*n)-a(n))/2 where b is the Inverse Euler transform of A000595. - Andrew Howroyd, Jan 27 2020

Extensions

a(0)=1 prepended and terms a(7) and beyond from Andrew Howroyd, Jan 26 2020
Previous Showing 51-53 of 53 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.