A245518 Irregular triangle read by rows: T(n,i) = number of alpha-labeled graphs with n edges that do not use the label i, for 1 <= i <= n-1 and n >= 4.

1, 0, 1, 4, 2, 2, 4, 16, 12, 8, 12, 16, 64, 64, 40, 40, 64, 64, 284, 328, 236, 176, 236, 328, 284, 1360, 1760, 1432, 1000, 1000, 1432, 1760, 1360, 7184, 9928, 9092, 6536, 5312, 6536, 9092, 9928, 7184

Offset: 4

Author

Sarah Minion and Christian Barrientos, Jul 24 2014

Keywords

• easy
• nonn
• tabf

Examples

			For n=4 and i=2, a(4,2) = 0.
For n=8 and i=5, a(8,5) = 64.
Triangle begins:
[n\i] [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]     [9]
[4]    1,      0,      1;
[5]    4,      2,      2,      4;
[6]    16,     12,     8,      12,     16;
[7]    64,     64,     40,     40,     64,     64;
[8]    284,    328,    236,    176,    236,    328,    284;
[9]    1360,   1760,   1432,   1000,   1000,   1432,   1760,   1360;
[10]   7184,   9928,   9092,   6536,   5312,   6536,   9092,   9928,   7184;
. . .
		

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,i) = sum_{L=1..^n-2} product_{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}.