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

1,3

Links

Table of n, a(n) for n=1..26.

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,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).

Example

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

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 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

Crossrefs

Cf. A320578, A320579.

Sequence in context: A170855 A338683 A361943 * A339317 A131814 A003620

Adjacent sequences: A320580 A320581 A320582 * A320584 A320585 A320586

Keyword

nonn,hard,tabf,more

Author

James Hammer, Daniel A. McGinnis, Oct 15 2018

Status

approved