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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.

%I A241094 #50 Jul 30 2024 09:20:56
%S A241094 0,1,1,4,4,4,18,24,24,18,96,144,144,96,600,960,1080,1080,960,600,4320,
%T A241094 7200,8460,8460,8460,7200,4320,35280,60840,75600,80640,80640,75600,
%U A241094 60480,35280,322560,564480,725760,806400,806400,806400,725760,564480,322560
%N 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.
%C A241094 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.
%H A241094 C. Barrientos and S. M. Minion, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Minion/minion3.pdf">Enumerating families of labeled graphs</a>, J. Integer Seq., 18(2015), article 15.1.7.
%H A241094 J. A. Gallian, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6">A dynamic survey of graph labeling</a>, Elec. J. Combin., (2013), #DS6.
%H A241094 David A. Sheppard, <a href="http://dx.doi.org/10.1016/0012-365X(76)90051-0">The factorial representation of major balanced labelled graphs</a>, Discrete Math., 15(1976), no. 4, 379-388.
%F A241094 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).
%F A241094 # alternative
%F A241094 A241094 := proc(n,i)
%F A241094     if n <2 or i<1 or i >= n then
%F A241094         0;
%F A241094     elif i <= floor(n/2) then
%F A241094         GAMMA(n-1)*(n-1-i)*i;
%F A241094     else
%F A241094         GAMMA(n-1)*(n-i)*(i-1) ;
%F A241094     fi ;
%F A241094 end proc:
%F A241094 seq(seq(A241094(n,i),i=1..n-1),n=2..12); # _R. J. Mathar_, Jul 30 2024
%e A241094 For n=7 and i=3, g(7,3) = 1080.
%e A241094 For n=7 and i=5, g(7,5) = 960.
%e A241094 Triangle begins:
%e A241094 [n\i]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
%e A241094 [2]     0;
%e A241094 [3]     1,      1;
%e A241094 [4]     4,      4,      4;
%e A241094 [5]    18,     24,     24,     18;
%e A241094 [6]    96,    144,    144,    144,     96;
%e A241094 [7]   600,    960,   1080,   1080,    960,    600;
%e A241094 [8]  4320,   7200,   8640,   8640,   8640,   7200,   4320;
%e A241094 [9] 35280,  60480,  75600,  80640,  80640,  75600,  60480,  35280;
%e A241094 ...
%e A241094 - _Bruno Berselli_, Apr 23 2014
%p A241094 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):
%t A241094 n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
%t A241094 For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
%t A241094 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]]]
%o A241094 (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
%Y A241094 Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.
%K A241094 nonn,tabl,easy
%O A241094 2,4
%A A241094 _Christian Barrientos_ and _Sarah Minion_, Apr 15 2014