A005193
a(n) is the number of alpha-labelings of graphs with n edges.
Original entry on oeis.org
1, 2, 4, 10, 30, 106, 426, 1930, 9690, 53578, 322650, 2106250, 14790810, 111327178, 893091930, 7614236170, 68695024410, 654301474378, 6557096219610, 69005893630090, 760519875693210, 8763511069234378, 105343011537811290, 1319139904954848010
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), #15.1.7.
- Henryk Fuks and Kate Sullivan, Enumeration of number-conserving cellular automata rules with two inputs, arXiv:0711.1349 [nlin.CG], 2007; Journal of Cellular Automata 2 vol. 2 pp. 141-148 (2007).
- David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.
-
A005193 := proc(q)
2*add((j!)^2*j^(q-2*j),j=1..q/2) ;
if type(q,'odd') then
%+((q+1)/2)!*((q-1)/2)! ;
else
% ;
end if;
end proc:
seq(A005193(n),n=1..40) ; # R. J. Mathar, Jul 13 2014
-
a[n_] := 2 Sum[(j!)^2*j^(n-2j), {j, 1, n/2}] + Boole[OddQ[n]]*((n+1)/2)! * ((n-1)/2)!;
Array[a, 24] (* Jean-François Alcover, Nov 20 2017 *)
A342357
Number of fundamentally different rainbow graceful labelings of graphs with n edges.
Original entry on oeis.org
1, 2, 11, 125, 1469, 30970, 1424807, 25646168, 943532049, 66190291008, 1883023236995, 119209289551407, 8338590851427689, 366451025462807402, 25231464507361789935, 2996947275258886238380, 211289282287835811874277, 12680220578500976681544666, 1815313698001596651227722787
Offset: 1
Each equivalence class has exactly one graph with v_1=0.
For n=3 the eleven classes of graphs 0v_2v_3 are: {000,011,015,050,054,065}, {001,002,024,041,063,064}, {003,026,031,034,046,062}, {004,061}, {005,013,021,044,052,060}, {006,014,030,035,051,066}, {010,055}, {012,020,022,043,045,053}, {016,025,032,033,040,056}, {023,042}, {036}.
- D. E. Knuth, The Art of Computer Programming, forthcoming exercise in Section 7.2.2.3.
- A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Dunod Paris (1967) 349-355.
-
sols[alf_,bet_,q_]:=Block[{d=GCD[alf,q]},If[Mod[bet,d]!=0,0,d]]
(* that many solutions to alf x == bet (modulo q) for 0<=xl && q-ll>l, s++;ll=Mod[ll*a,q];r=Mod[r*a+1,q]];
If[ll==l,sols[a^s-1,-r b,q], If[q-ll==l,sols[a^s-1,l-r b,q],1]]]
f[a_,b_,q_]:=Product[f[l,a,b,q],{l,(q-1)/2}]
x[q_]:=Sum[If[GCD[a,q]>1,0,Sum[f[a,b,q],{b,0,q-1}]],{a,q-1}]/(q EulerPhi[q])
a[n_]:=x[2n+1]
-
# This is a port of the Mathematica program.
def sols(a, b, q):
g = gcd(a, q)
return 0 if mod(b, g) != 0 else g
def F(k, a, b, q):
s, r, m = 1, 1, mod(k*a, q)
while m > k and q - m > k:
s += 1
m = mod(m*a, q)
r = mod(r*a + 1, q)
if m == k: return sols(a^s - 1, -r*b, q)
if m == q-k: return sols(a^s - 1, k - r*b, q)
return 1
def f(a, b, q):
return prod(F(k, a, b, q) for k in (1..(q-1)//2))
def a(n):
q = 2*n + 1
s = sum(0 if gcd(a, q) > 1 else sum(f(a, b, q)
for b in (0..q-1)) for a in (1..q-1))
return s // (q*euler_phi(q))
print([a(n) for n in (1..19)]) # Peter Luschny, Mar 10 2021
Showing 1-2 of 2 results.
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