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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).
0
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
4,3
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.
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}.
EXAMPLE
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;
...
CROSSREFS
Sequence in context: A170897 A217771 A261321 * A179526 A098525 A141666
KEYWORD
nonn,tabf,easy
AUTHOR
STATUS
approved