A241094 -id:A241094 - cp's OEIS frontend

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

Sarah Minion and Christian Barrientos, Jul 24 2014

			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;
...

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.

Cf. A241094, A005193.

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}.

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.

Original entry on oeis.org

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

Sarah Minion and Christian Barrientos, Jul 24 2014

			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;
. . .

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.

Cf. A241094, A005193.

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}.

A245519 Number of alpha-labeled graphs with n edges and at most n vertices.

Original entry on oeis.org

0, 0, 0, 2, 10, 64, 336, 1872, 11104, 71944, 508032, 3511232, 27192704, 223750464, 1947253504, 17899536448, 173156535168, 1760383827776, 18752453106176, 209034916385472, 2432351796434560, 29509268795249700

Offset: 1

Sarah Minion and Christian Barrientos, Jul 24 2014

			For n=4, a(4)=2, there are 2 alpha-labeled graphs with 4 edges and at most 4 vertices.
For n=10, a(10)=71944, there are 71944 alpha-labeled graphs with 10 edges and at most 10 vertices.

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.

Cf. A241094, A005193.

a(n) = Sum_{L=1..n-2} Sum_{i=1..n-1} 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}.

cp's OEIS Frontend

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A245519 Number of alpha-labeled graphs with n edges and at most n vertices.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula