A320578 Triangle read by rows: T(n,k) is the number of permutation graphs on n vertices with domination number k, with 1 <= k <= n.

1, 1, 1, 3, 2, 1, 10, 10, 3, 1, 43, 54, 18, 4, 1, 223, 351, 113, 27, 5, 1, 1364, 2613, 833, 186, 37, 6, 1, 9643, 21965, 6921, 1461, 274, 48, 7, 1, 77545, 205780, 64128, 12727, 2253, 378, 60, 8, 1, 699954, 2127068, 655391, 122345, 20230, 3230, 499, 73, 9, 1

Offset

1,4

Links

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

Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.

Formula

T(n,k) = A320579(n,k) + A320583(n,k).

T(n,1) = A320583(n,1).

Example

Triangle begins:
    1;
    1,   1;
    3,   2,   1;
   10,  10,   3,   1;
   43,  54,  18,   4,   1;
  223, 351, 113,  27,   5,   1;
  ...

Prog

(Python)
import networkx as nx
import math
def permutation(lst):
    if len(lst) == 0:
        return []
    if len(lst) == 1:
        return [lst]
    l = []
    for i in range(len(lst)):
        m = lst[i]
        remLst = lst[:i] + lst[i + 1:]
        for p in permutation(remLst):
            l.append([m] + p)
    return l
def generatePermsOfSizeN(n):
    lst = []
    for i in range(n):
        lst.append(i+1)
    return permutation(lst)
def powersetHelper(A):
    if A == []:
        return [[]]
    a = A[0]
    incomplete_pset = powersetHelper(A[1:])
    rest = []
    for set in incomplete_pset:
        rest.append([a] + set)
    return rest + incomplete_pset
def powerset(A):
    ps = powersetHelper(A)
    ps.sort(key = len)
    return ps
    print(ps)
def countDomNumbersOnN(n):
    lst=[]
    perms = generatePermsOfSizeN(n)
    for i in range(n):
        lst.append(i+1)
    ps = powerset(lst)
    dic={}
    for perm in perms:
        tempGraph = nx.Graph()
        tempGraph.add_nodes_from(perm)
        for i in range(len(perm)):
            for k in range(i+1, len(perm)):
                if perm[k] < perm[i]:
                    tempGraph.add_edge(perm[i], perm[k])
        for p in ps:
            if nx.is_dominating_set(tempGraph, p):
                dom = len(p)
                if dom in dic:
                    dic[dom] += 1
                    break
                else:
                    dic[dom] = 1
                    break
    return dic

Crossrefs

Cf. A320579, A320583

Sequence in context: A116071 A214622 A327801 * A267836 A319669 A325305

Adjacent sequences: A320575 A320576 A320577 * A320579 A320580 A320581

Keyword

nonn,hard,tabl

Author

James Hammer, Daniel A. McGinnis, Oct 15 2018

Status

approved