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A245517 Irregular triangle read by rows: T(n,L) = number of alpha-labeled graphs with n edges and boundary value L that do not use one number from (1,2,...,n-1) as a label (n >= 4, 1 <= L <= n - 2).

%I A245517 #20 Nov 22 2017 02:02:39
%S A245517 1,1,4,4,4,12,20,20,12,32,88,96,88,32,80,352,504,504,352,80,192,1328,
%T A245517 2592,2880,2592,1328,192,448,4816,12852,17280,17280,12852,4816,448
%N A245517 Irregular triangle read by rows: T(n,L) = number of alpha-labeled graphs with n edges and boundary value L that do not use one number from (1,2,...,n-1) as a label (n >= 4, 1 <= L <= n - 2).
%H A245517 Christian Barrientos, Sarah Minion, <a href="https://dx.doi.org/10.7151/dmgt.1985">On the number of alpha-labeled graphs</a>, Discussiones Mathematicae Graph Theory, to appear.
%H A245517 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 A245517 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 A245517 a(n,L,i) = \sum_{i = 1}^{n - 1} \prod_{k = 1}^{n} d(L,k,i), where
%F A245517 for i < L,
%F A245517             d(L,k)       if 1 <= k <= i,
%F A245517 d(L,k,i) ={ d(L,k) - 1   if i < k < n - i,
%F A245517             d(L,k)       if n - i <= k <= n;
%F A245517 for i > L + 1,
%F A245517             d(L,k)       if 1 <= k <= n - i,
%F A245517 d(L,k,i) ={ d(L,k) - 1   if n - i < k < n - i + L + 2,
%F A245517             d(L,k)       if n - i + L + 2 <= k <= n.
%F A245517           k          if 1 <= k < m,
%F A245517 d(L,k) ={ L + 1      if m <= k <= M,
%F A245517           n + 1 - k  if M < k <= n,
%F A245517 m = min{L + 1, n - L}, M = max{L + 1, n - L}.
%e A245517 For n=9 and L=5, T(9,5) = 2592.
%e A245517 For n=10 and L=4, T(10,4) = 17280.
%e A245517 Triangle begins:
%e A245517 [n\L]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
%e A245517 [4]     1,      1;
%e A245517 [5]     4,      4,      4;
%e A245517 [6]     12,     20,     20,     12;
%e A245517 [7]     32,     88,     96,     88,     32;
%e A245517 [8]     80,     352,    504,    504,    352,    80;
%e A245517 [9]     192,    1328,   2592,   2880,   2592,   1328,   192;
%e A245517 [10]    448,    4816,   12852,  17280,  17280,  12852,  4816,   448;
%e A245517 ...
%Y A245517 Cf. A241094, A005193.
%K A245517 nonn,tabf,easy
%O A245517 4,3
%A A245517 _Sarah Minion_ and _Christian Barrientos_, Jul 24 2014