A320583 Irregular triangle read by rows: T(n,k) is the number of connected permutation graphs on n vertices with domination number k, with 1 <= k <= floor(n/2).

1, 1, 3, 10, 3, 43, 28, 223, 236, 2, 1364, 1842, 62, 9643, 18433, 1015, 2, 77545, 181568, 14146, 84, 699954, 1938199, 189077, 2093, 2

Offset: 1

Author

James Hammer, Daniel A. McGinnis, Oct 15 2018

Keywords

• nonn
• hard
• tabf
• more

Examples

			Triangle begins:
    1;
    1;
    3;
   10,   3;
   43,  28;
  223, 236,  2;
  ...
		

Links

• 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.

Crossrefs

Cf. A320578, A320579.

Programs

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 countcnctdDomNumbersOnN(n):
    lst=[]
    l=[]
    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])
        if nx.is_connected(tempGraph)==True:
            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

Formula

T(n,n/2) = 2 for even n. See Theorem 4.5 in the link by Theresa Baren, et al.

T(n,k) = A320578(n,k) - A320579(n,k).

T(n,1) = A320578(n,1).