A320579 Triangle read by rows: T(n,k) is the number of disconnected permutation graphs on n vertices with domination number k, with 2 <= k <= n.

1, 2, 1, 7, 3, 1, 26, 18, 4, 1, 115, 111, 27, 5, 1, 592, 771, 186, 37, 6, 1, 3532, 5906, 1459, 274, 48, 7, 1, 24212, 49982, 12643, 2253, 378, 60, 8, 1, 188869, 466314, 120252, 20228, 3230, 499, 73, 9, 1

Offset

2,2

Links

Table of n, a(n) for n=2..46.

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) = A320578(n,k) - A320583(n,k).

Example

Triangle begins:
    1;
    2,   1;
    7,   3,   1;
   26,  18,   4,  1;
  115, 111,  27,  5,  1;
  592, 771, 186, 37,  6,  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 countdisDomNumbersOnN(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)==False:
            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, A320583.

Sequence in context: A197328 A352247 A136535 * A091370 A125697 A090699

Adjacent sequences: A320576 A320577 A320578 * A320580 A320581 A320582

Keyword

nonn,tabl,hard,more

Author

James Hammer, Daniel A. McGinnis, Oct 15 2018

Status

approved