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A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L. 0
2, 4, 8, 8, 32, 48, 16, 128, 288, 384, 32, 512, 1728, 3072, 3840, 64, 2048, 10368, 24576, 38400, 46080, 128, 8192, 62208, 196608, 384000, 552960, 645120, 256, 32768, 373248, 1572864, 3840000, 6635520, 9031680, 10321920, 512, 131072, 2239488, 12582912, 38400000, 79626240, 126443520, 165150720, 185794560, 1024, 524288, 13436928, 100663296, 384000000, 955514880, 1770209280, 2642411520, 3344302080, 3715891200 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A graph with n edges is rho-labeled if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as label the absolute difference of its end-vertices and the edge labels are x_1, x_2, ..., x_n where x_i = i or x_i = 2n + 1 - i. A rho-labeling of a bipartite graph is said to be bipartite when the labels of one stable set are smaller than the labels on the other stable set. The largest of the smaller vertex labels is its boundary value.

From Robert G. Wilson v, Jul 05 2015: (Start)

The columns:

T(n, 0) = 2^n,

T(n, 1) = 2^(2n-1),

T(n, 2) = 2^(n+1)*3^(n-2),

T(n, 3) = 3*2^(3n-5),

T(n, 4) = 3*2^(n+3)*5^(n-4),

T(n, 5) = 5*2^(2n-2)*3^(n-4), etc.

The diagonals:

the main,            T(n, n-1) = 2^n*n*(n-1!) = 2*A002866,

the second diagonal, T(n, n-2) = 2^n*(n-1)^2*(n-2)! = 4*A014479,

the third diagonal,  T(n, n-3) = 2^n*(n-2)^3*(n-3)!,

the k_th diagonal,   T(n, n-k) = 2^n*(n-k)^k*(n-k)!, etc.

... (End)

LINKS

Table of n, a(n) for n=1..55.

Joseph A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2014), #DS6.

FORMULA

For n>=1, 0<=L<=n-1, T(n,L)=(2^n)*(L!)*(L+1)^(n-L).

EXAMPLE

For n=5 and L=1, T(5,1)=(2^5)*(1!)*(1+1)^(5-1)=512.

For n=9 and L=3, T(9,3)=12582912.

Triangle, T, begins:

-----------------------------------------------------------------------------

n\L |   0       1         2          3          4          5           6

----|------------------------------------------------------------------------

1   |   2;

2   |   4,      8;

3   |   8,     32,       48;

4   |  16,    128,      288,       384;

5   |  32,    512,     1728,      3072,      3840;

6   |  64,   2048,    10368,     24576,     38400,     46080;

7   | 128,   8192,    62208,    196608,    384000,    552960,     645120;

8   | 256,  32768,   373248,   1572864,   3840000,   6635520,    9031680, ...

...

For n=2 and L=1, T(2,1)=8, because: the bipartite graph <{v1,v2,v3},{x1=v1v2,x2=v1v3}> has rho-labelings (2,1,3),(2,1,4) with L=1 on the stable set {x2} and rho-labelings (1,2,0),(0,4,1) with L=1 on the stable set {x1,x3}; the bipartite graph <{v1,v2,v3,v4},{x1=v1v2,x2=v3v4}> has rho-labeling (0,4,1,3),(1,2,0,3) with L=1 on the stable set {v1,v3} and rho-labeling (4,0,3,1),(2,1,3,0) with L=1 on the stable set {v2,v4}. - Danny Rorabaugh, Apr 03 2015

MATHEMATICA

t[n_, l_] := 2^n*l!(l+1)^(n-l); Table[ t[n, l], {n, 8}, {l, 0, n-1}] // Flatten (* Robert G. Wilson v, Jul 05 2015 *)

PROG

(MAGMA) [2^n*Factorial(l)*(l+1)^(n-l): l in [0..n-1], n in [1..10]]; // Bruno Berselli, Aug 05 2015

CROSSREFS

Sequence in context: A140119 A273068 A193846 * A011402 A131625 A196000

Adjacent sequences:  A255905 A255906 A255907 * A255909 A255910 A255911

KEYWORD

easy,nonn,tabl

AUTHOR

Christian Barrientos and Sarah Minion, Mar 10 2015

STATUS

approved

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Last modified August 6 19:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)