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).

1, 1, 4, 4, 4, 12, 20, 20, 12, 32, 88, 96, 88, 32, 80, 352, 504, 504, 352, 80, 192, 1328, 2592, 2880, 2592, 1328, 192, 448, 4816, 12852, 17280, 17280, 12852, 4816, 448

Offset: 4

Author

Sarah Minion and Christian Barrientos, Jul 24 2014

Keywords

• nonn
• tabf
• easy

Examples

			For n=9 and L=5, T(9,5) = 2592.
For n=10 and L=4, T(10,4) = 17280.
Triangle begins:
[n\L]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[4]     1,      1;
[5]     4,      4,      4;
[6]     12,     20,     20,     12;
[7]     32,     88,     96,     88,     32;
[8]     80,     352,    504,    504,    352,    80;
[9]     192,    1328,   2592,   2880,   2592,   1328,   192;
[10]    448,    4816,   12852,  17280,  17280,  12852,  4816,   448;
...
		

Links

• Christian Barrientos, Sarah Minion, On the number of alpha-labeled graphs, Discussiones Mathematicae Graph Theory, to appear.
• 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. A241094, A005193.

Formula

a(n,L,i) = \sum_{i = 1}^{n - 1} \prod_{k = 1}^{n} d(L,k,i), where

for i < L,

d(L,k) if 1 <= k <= i,

d(L,k,i) ={ d(L,k) - 1 if i < k < n - i,

d(L,k) if n - i <= k <= n;

for i > L + 1,

d(L,k) if 1 <= k <= n - i,

d(L,k,i) ={ d(L,k) - 1 if n - i < k < n - i + L + 2,

d(L,k) if n - i + L + 2 <= k <= n.

k if 1 <= k < m,

d(L,k) ={ L + 1 if m <= k <= M,

n + 1 - k if M < k <= n,

m = min{L + 1, n - L}, M = max{L + 1, n - L}.