A241094 Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.

0, 1, 1, 4, 4, 4, 18, 24, 24, 18, 96, 144, 144, 96, 600, 960, 1080, 1080, 960, 600, 4320, 7200, 8460, 8460, 8460, 7200, 4320, 35280, 60840, 75600, 80640, 80640, 75600, 60480, 35280, 322560, 564480, 725760, 806400, 806400, 806400, 725760, 564480, 322560

Offset: 2

Author

Christian Barrientos and Sarah Minion, Apr 15 2014

Keywords

• nonn
• tabl
• easy

Comments

A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n.

Examples

			For n=7 and i=3, g(7,3) = 1080.
For n=7 and i=5, g(7,5) = 960.
Triangle begins:
[n\i]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]     0;
[3]     1,      1;
[4]     4,      4,      4;
[5]    18,     24,     24,     18;
[6]    96,    144,    144,    144,     96;
[7]   600,    960,   1080,   1080,    960,    600;
[8]  4320,   7200,   8640,   8640,   8640,   7200,   4320;
[9] 35280,  60480,  75600,  80640,  80640,  75600,  60480,  35280;
...
- _Bruno Berselli_, Apr 23 2014
		

Links

• C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), article 15.1.7.
• J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
• David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Crossrefs

Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.

Programs

Magma

/* As triangle: */ [[i le Floor(n/2) select Factorial(n-2)*(n-1-i)*i else Factorial(n-2)*(n-i)*(i-1): i in [1..n-1]]: n in [2..10]]; // Bruno Berselli, Apr 23 2014

Maple

Labeled:=(i,n) piecewise(n<2 or i<1, -infinity, 1 <= i <= floor(n/2), GAMMA(n-1)*(n-1-i)*i, ceil((n+1)/2) <= i <= n-1, GAMMA(n-1)*(n-i)*(i-1), infinity):

Mathematica

n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
For[i = Ceiling[(n+1)/2], i <= (n-1), i++, g[n_,i_]:=(n-2)!*(n-i)*(i-1); Print["g(",n,",",i,")=",g[n,i]]]

Formula

For n >=2, if 1 <= i <= floor(n/2), g(n,i) = (n-2)!*(n-1-i)*i; if ceiling((n+1)/2) <= i <= n-1, g(n,i) = (n-2)!*(n-i)*(i-1).

# alternative

A241094 := proc(n,i)

if n <2 or i<1 or i >= n then

0;

elif i <= floor(n/2) then

GAMMA(n-1)*(n-1-i)*i;

else

GAMMA(n-1)*(n-i)*(i-1) ;

fi ;

end proc:

seq(seq(A241094(n,i),i=1..n-1),n=2..12); # R. J. Mathar, Jul 30 2024