note: all software on this page released under the MIT license
update 2017-05-18: an optimized version of the source is here.
update 2017-05-20: a parallel optimized version of the source in solution format is here. You can compile with mono from the command line with
mcs -platform:x64 -r:System.Numerics Program.cs
Imagine a question like, "Given a complete graph with n nodes, how many cliques of size k are there?"
I want to discuss two different specifics of this question, one with rotation, and the other with rotation and reflection.
The first case, up to isomorphism, is easily answered with the binomial coefficient, and given by C(n,k).
Now how do you count this? A recusive program is given in this stackoverflow answer.
Alternatively, there is also an iterative solution. Each iteration essentially asks, "what is the next smallest sum of k distinct powers of 2?" Or in binary, "what is the next number with only k bits set?"
Here is an algorithm I wrote in C#. A graph is represented by an array of size n; setting a value of the array is equivalent to choosing a node. For example, [1,1,0,1,0] is a graph of five nodes, with three nodes chosen for a clique.
// returns true if an increment occurred, otherwise returns false.
// modifies the input array in place.
// This function is named "Increment" but there are several noted places in code that
// can swap "0" and "1" to change this to decrement.
public static bool Increment(int[] arr)
{
// There are two cases, the easy one and the hard one.
// The easy case just shifts the first unset digit right.
// The hard case is when the first unset digit is in the rightmost spot.
/*
example array of length 6 with 3 digits set, same as C(6,3)
5 4 3 2 1 0 // index position in array
---------------------
1 1 1 0 0 0 // Increment stops here
1 1 0 1 0 0
1 1 0 0 1 0
1 1 0 0 0 1
1 0 1 1 0 0
1 0 1 0 1 0
1 0 1 0 0 1
1 0 0 1 1 0
1 0 0 1 0 1
1 0 0 0 1 1
0 1 1 1 0 0
0 1 1 0 1 0
0 1 1 0 0 1
0 1 0 1 1 0
0 1 0 1 0 1
0 1 0 0 1 1
0 0 1 1 1 0
0 0 1 1 0 1
0 0 1 0 1 1
0 0 0 1 1 1 // Increment starts here
*/
int startPos = 0;
int carry = 0;
int index;
// find the first digit that's set
for (index = 0; index < arr.Length; index++)
{
if (arr[index] == 0) // toggle to change direction
{
startPos = index;
break;
}
}
// Easy case. The digit can be shifted.
if (startPos > 0)
{
arr[startPos] = 1; // toggle to change direction
arr[startPos - 1] = 0; // toggle to change direction
return true;
}
// Else, the hard case. Flip and count the digits that need to be carried.
for (index = startPos; index < arr.Length; index++)
{
if (arr[index] == 0) // toggle to change direction
{
arr[index] = 1; // toggle to change direction
carry++;
}
else
{
break;
}
}
// Travel down the array until it's possible to carry.
for (index = carry + 1; index < arr.Length; index++)
{
if (arr[index] == 0) // toggle to change direction
{
startPos = index;
break;
}
}
// If the end of the array was reached, it's not possible to
// do anything. (max value for increment or min value for decrement)
if (index >= arr.Length)
{
return false;
}
// Flip the digits "supplying" the carry
arr[startPos] = 1; // toggle to change direction
arr[startPos - 1] = 0; // toggle to change direction
// Flip the digits that are carried down.
for (index = startPos - 2; carry > 0; carry--, index--)
{
arr[index] = 0; // toggle to change direction
}
return true;
}
The total number of times an array can be incremented from the starting value should be the same as the binomial coefficient.
In order to answer the initial question, it's necessary to discount isomorphic graphs. That is, [1,1,1,0,0] and [1,1,0,0,1] and [0,1,1,1,0] are isomorphic and shouldn't be counted more than once.
I handle this by generating a "prime" id for each graph. An id is generated by treating each index of the array as a bit; the array is then interpretted as a binary integer. A "prime" id rotates the array through one full cycle and returns the minimum id that was found for each iteration of the rotation.
When an id is first encountered, it is stored. Each rotation of the array is also stored (as its id) in a collection called a family. If an id has been encountered before, this iteration of C(n,k) is skipped.
Here's my implementation in C#
// returns all possible families. A family contains a list of
// isomorphic graphs, as defined by their prime id.
public static List<List<UInt64>> FamilyCount(int nodes, int cliqueSize)
{
var nodeList = new int[nodes];
var families = new List<List<UInt64>>();
int currentFamily = 0;
// is dictionary the right data structure since the only thing
// that matters is whether the key exists? The value isn't read ...
var seen = new Dictionary<UInt64, int>();
for (int i = 0; i < cliqueSize; i++)
{
nodeList[i] = 1;
}
do
{
currentFamily = -1;
// the array will be modified, so leave the original unchanged.
var copy = (new List<int>(nodeList)).ToArray();
// will contain list of all ids found
var iterationFamily = new List<UInt64>();
// get prime id to determine if that has been seen before.
var id = GetId(copy);
if (seen.ContainsKey(id) )
{
continue;
}
iterationFamily.Add(id);
// this hasn't been seen before, so it must be a new family
currentFamily = families.Count;
families.Add(new List<UInt64>());
seen.Add(id, currentFamily);
for (int i = 0; i < nodes; i++)
{
RotateRight(copy);
id = AsInt(copy);
iterationFamily.Add(id);
}
iterationFamily.Sort();
foreach (var item in iterationFamily)
{
if (!families[currentFamily].Contains(item))
{
families[currentFamily].Add(item);
}
}
} while (Increment(nodeList));
return families;
}
This is enough to build a program that iterates through C(n,k) up to n=20. You can find the complete source here
Results. Note that results are symmetric and the table is not extended for brevity.
| F(x,1) | F(x,2) | F(x,3) | F(x,4) | F(x,5) | F(x,6) | F(x,7) | F(x,8) | F(x,9) | F(x,10) | |
| F(2,x) | 1 | |||||||||
| F(3,x) | 1 | 1 | ||||||||
| F(4,x) | 1 | 2 | 1 | |||||||
| F(5,x) | 1 | 2 | 2 | 1 | ||||||
| F(6,x) | 1 | 3 | 4 | 3 | 1 | |||||
| F(7,x) | 1 | 3 | 5 | 5 | 3 | 1 | ||||
| F(8,x) | 1 | 4 | 7 | 10 | 7 | 4 | 1 | |||
| F(9,x) | 1 | 4 | 10 | 14 | 14 | 10 | 4 | 1 | ||
| F(10,x) | 1 | 5 | 12 | 22 | 26 | 22 | 12 | 5 | 1 | |
| F(11,x) | 1 | 5 | 15 | 30 | 42 | 42 | 30 | 15 | 5 | 1 |
| F(12,x) | 1 | 6 | 19 | 43 | 66 | 80 | 66 | 43 | 19 | 6 |
| F(13,x) | 1 | 6 | 22 | 55 | 99 | 132 | 132 | 99 | 55 | 22 |
| F(14,x) | 1 | 7 | 26 | 73 | 143 | 217 | 246 | 217 | 143 | 73 |
| F(15,x) | 1 | 7 | 31 | 91 | 201 | 335 | 429 | 429 | 335 | 201 |
| F(16,x) | 1 | 8 | 35 | 116 | 273 | 504 | 715 | 810 | 715 | 504 |
| F(17,x) | 1 | 8 | 40 | 140 | 364 | 728 | 1144 | 1430 | 1430 | 1144 |
| F(18,x) | 1 | 9 | 46 | 172 | 476 | 1038 | 1768 | 2438 | 2704 | 2438 |
| F(19,x) | 1 | 9 | 51 | 204 | 612 | 1428 | 2652 | 3978 | 4862 | 4862 |
This is also th enumber of necklaces with k black beads and n-k white beads. See http://oeis.org/A047996.
Full output for n=2..40 in csv format is available here.
The program also counts the number of ids in each family. For instance, [0,1,0,1,0,1] generates a family with only two ids, even though it will be rotated six times. The statistics gathered are simply [number of ids by number of families with that many ids]. Summing the rightmost values in each row will give the same number as above. When n and k are coprime there is only one size of family. If n is prime, all families have the same number of ids.
FamilyCount( 4, 1): [ 4x 1] FamilyCount( 4, 2): [ 2x 1] [ 4x 1] FamilyCount( 4, 3): [ 4x 1] FamilyCount( 5, 1): [ 5x 1] FamilyCount( 5, 2): [ 5x 2] FamilyCount( 5, 3): [ 5x 2] FamilyCount( 5, 4): [ 5x 1] FamilyCount( 6, 1): [ 6x 1] FamilyCount( 6, 2): [ 3x 1] [ 6x 2] FamilyCount( 6, 3): [ 2x 1] [ 6x 3] FamilyCount( 6, 4): [ 3x 1] [ 6x 2] FamilyCount( 6, 5): [ 6x 1] FamilyCount( 7, 1): [ 7x 1] FamilyCount( 7, 2): [ 7x 3] FamilyCount( 7, 3): [ 7x 5] FamilyCount( 7, 4): [ 7x 5] FamilyCount( 7, 5): [ 7x 3] FamilyCount( 7, 6): [ 7x 1] FamilyCount( 8, 1): [ 8x 1] FamilyCount( 8, 2): [ 4x 1] [ 8x 3] FamilyCount( 8, 3): [ 8x 7] FamilyCount( 8, 4): [ 2x 1] [ 4x 1] [ 8x 8] FamilyCount( 8, 5): [ 8x 7] FamilyCount( 8, 6): [ 4x 1] [ 8x 3] FamilyCount( 8, 7): [ 8x 1] FamilyCount( 9, 1): [ 9x 1] FamilyCount( 9, 2): [ 9x 4] FamilyCount( 9, 3): [ 3x 1] [ 9x 9] FamilyCount( 9, 4): [ 9x 14] FamilyCount( 9, 5): [ 9x 14] FamilyCount( 9, 6): [ 3x 1] [ 9x 9] FamilyCount( 9, 7): [ 9x 4] FamilyCount( 9, 8): [ 9x 1] FamilyCount( 10, 1): [ 10x 1] FamilyCount( 10, 2): [ 5x 1] [ 10x 4] FamilyCount( 10, 3): [ 10x 12] FamilyCount( 10, 4): [ 5x 2] [ 10x 20] FamilyCount( 10, 5): [ 2x 1] [ 10x 25] FamilyCount( 10, 6): [ 5x 2] [ 10x 20] FamilyCount( 10, 7): [ 10x 12] FamilyCount( 10, 8): [ 5x 1] [ 10x 4] FamilyCount( 10, 9): [ 10x 1] FamilyCount( 11, 1): [ 11x 1] FamilyCount( 11, 2): [ 11x 5] FamilyCount( 11, 3): [ 11x 15] FamilyCount( 11, 4): [ 11x 30] FamilyCount( 11, 5): [ 11x 42] FamilyCount( 11, 6): [ 11x 42] FamilyCount( 11, 7): [ 11x 30] FamilyCount( 11, 8): [ 11x 15] FamilyCount( 11, 9): [ 11x 5] FamilyCount( 11, 10): [ 11x 1] FamilyCount( 12, 1): [ 12x 1] FamilyCount( 12, 2): [ 6x 1] [ 12x 5] FamilyCount( 12, 3): [ 4x 1] [ 12x 18] FamilyCount( 12, 4): [ 3x 1] [ 6x 2] [ 12x 40] FamilyCount( 12, 5): [ 12x 66] FamilyCount( 12, 6): [ 2x 1] [ 4x 1] [ 6x 3] [ 12x 75] FamilyCount( 12, 7): [ 12x 66] FamilyCount( 12, 8): [ 3x 1] [ 6x 2] [ 12x 40] FamilyCount( 12, 9): [ 4x 1] [ 12x 18] FamilyCount( 12, 10): [ 6x 1] [ 12x 5] FamilyCount( 12, 11): [ 12x 1] FamilyCount( 13, 1): [ 13x 1] FamilyCount( 13, 2): [ 13x 6] FamilyCount( 13, 3): [ 13x 22] FamilyCount( 13, 4): [ 13x 55] FamilyCount( 13, 5): [ 13x 99] FamilyCount( 13, 6): [ 13x 132] FamilyCount( 13, 7): [ 13x 132] FamilyCount( 13, 8): [ 13x 99] FamilyCount( 13, 9): [ 13x 55] FamilyCount( 13, 10): [ 13x 22] FamilyCount( 13, 11): [ 13x 6] FamilyCount( 13, 12): [ 13x 1] FamilyCount( 14, 1): [ 14x 1] FamilyCount( 14, 2): [ 7x 1] [ 14x 6] FamilyCount( 14, 3): [ 14x 26] FamilyCount( 14, 4): [ 7x 3] [ 14x 70] FamilyCount( 14, 5): [ 14x 143] FamilyCount( 14, 6): [ 7x 5] [ 14x 212] FamilyCount( 14, 7): [ 2x 1] [ 14x 245] FamilyCount( 14, 8): [ 7x 5] [ 14x 212] FamilyCount( 14, 9): [ 14x 143] FamilyCount( 14, 10): [ 7x 3] [ 14x 70] FamilyCount( 14, 11): [ 14x 26] FamilyCount( 14, 12): [ 7x 1] [ 14x 6] FamilyCount( 14, 13): [ 14x 1] FamilyCount( 15, 1): [ 15x 1] FamilyCount( 15, 2): [ 15x 7] FamilyCount( 15, 3): [ 5x 1] [ 15x 30] FamilyCount( 15, 4): [ 15x 91] FamilyCount( 15, 5): [ 3x 1] [ 15x 200] FamilyCount( 15, 6): [ 5x 2] [ 15x 333] FamilyCount( 15, 7): [ 15x 429] FamilyCount( 15, 8): [ 15x 429] FamilyCount( 15, 9): [ 5x 2] [ 15x 333] FamilyCount( 15, 10): [ 3x 1] [ 15x 200] FamilyCount( 15, 11): [ 15x 91] FamilyCount( 15, 12): [ 5x 1] [ 15x 30] FamilyCount( 15, 13): [ 15x 7] FamilyCount( 15, 14): [ 15x 1] FamilyCount( 16, 1): [ 16x 1] FamilyCount( 16, 2): [ 8x 1] [ 16x 7] FamilyCount( 16, 3): [ 16x 35] FamilyCount( 16, 4): [ 4x 1] [ 8x 3] [ 16x 112] FamilyCount( 16, 5): [ 16x 273] FamilyCount( 16, 6): [ 8x 7] [ 16x 497] FamilyCount( 16, 7): [ 16x 715] FamilyCount( 16, 8): [ 2x 1] [ 4x 1] [ 8x 8] [ 16x 800] FamilyCount( 16, 9): [ 16x 715] FamilyCount( 16, 10): [ 8x 7] [ 16x 497] FamilyCount( 16, 11): [ 16x 273] FamilyCount( 16, 12): [ 4x 1] [ 8x 3] [ 16x 112] FamilyCount( 16, 13): [ 16x 35] FamilyCount( 16, 14): [ 8x 1] [ 16x 7] FamilyCount( 16, 15): [ 16x 1] FamilyCount( 17, 1): [ 17x 1] FamilyCount( 17, 2): [ 17x 8] FamilyCount( 17, 3): [ 17x 40] FamilyCount( 17, 4): [ 17x 140] FamilyCount( 17, 5): [ 17x 364] FamilyCount( 17, 6): [ 17x 728] FamilyCount( 17, 7): [ 17x 1144] FamilyCount( 17, 8): [ 17x 1430] FamilyCount( 17, 9): [ 17x 1430] FamilyCount( 17, 10): [ 17x 1144] FamilyCount( 17, 11): [ 17x 728] FamilyCount( 17, 12): [ 17x 364] FamilyCount( 17, 13): [ 17x 140] FamilyCount( 17, 14): [ 17x 40] FamilyCount( 17, 15): [ 17x 8] FamilyCount( 17, 16): [ 17x 1] FamilyCount( 18, 1): [ 18x 1] FamilyCount( 18, 2): [ 9x 1] [ 18x 8] FamilyCount( 18, 3): [ 6x 1] [ 18x 45] FamilyCount( 18, 4): [ 9x 4] [ 18x 168] FamilyCount( 18, 5): [ 18x 476] FamilyCount( 18, 6): [ 3x 1] [ 6x 2] [ 9x 9] [ 18x 1026] FamilyCount( 18, 7): [ 18x 1768] FamilyCount( 18, 8): [ 9x 14] [ 18x 2424] FamilyCount( 18, 9): [ 2x 1] [ 6x 3] [ 18x 2700] FamilyCount( 18, 10): [ 9x 14] [ 18x 2424] FamilyCount( 18, 11): [ 18x 1768] FamilyCount( 18, 12): [ 3x 1] [ 6x 2] [ 9x 9] [ 18x 1026] FamilyCount( 18, 13): [ 18x 476] FamilyCount( 18, 14): [ 9x 4] [ 18x 168] FamilyCount( 18, 15): [ 6x 1] [ 18x 45] FamilyCount( 18, 16): [ 9x 1] [ 18x 8] FamilyCount( 18, 17): [ 18x 1] FamilyCount( 19, 1): [ 19x 1] FamilyCount( 19, 2): [ 19x 9] FamilyCount( 19, 3): [ 19x 51] FamilyCount( 19, 4): [ 19x 204] FamilyCount( 19, 5): [ 19x 612] FamilyCount( 19, 6): [ 19x 1428] FamilyCount( 19, 7): [ 19x 2652] FamilyCount( 19, 8): [ 19x 3978] FamilyCount( 19, 9): [ 19x 4862] FamilyCount( 19, 10): [ 19x 4862] FamilyCount( 19, 11): [ 19x 3978] FamilyCount( 19, 12): [ 19x 2652] FamilyCount( 19, 13): [ 19x 1428] FamilyCount( 19, 14): [ 19x 612] FamilyCount( 19, 15): [ 19x 204] FamilyCount( 19, 16): [ 19x 51] FamilyCount( 19, 17): [ 19x 9] FamilyCount( 19, 18): [ 19x 1]
Now for the other part of the question -- what if a graph reflection is treated the same as a rotation? The above program is simply modified to also reverse the array and look for its prime id as well. While iterating through possible cliques, each rotation is also reversed. These are then treated as being in the same family. You can find the complete source code here. Run time is about twice as long as above.
Results. Note that results are symmetric and the table is not extended for brevity.
| F(x,1) | F(x,2) | F(x,3) | F(x,4) | F(x,5) | F(x,6) | F(x,7) | F(x,8) | F(x,9) | F(x,10) | |
| F(2,x) | 1 | |||||||||
| F(3,x) | 1 | 1 | ||||||||
| F(4,x) | 1 | 2 | 1 | |||||||
| F(5,x) | 1 | 2 | 2 | 1 | ||||||
| F(6,x) | 1 | 3 | 3 | 3 | 1 | |||||
| F(7,x) | 1 | 3 | 4 | 4 | 3 | 1 | ||||
| F(8,x) | 1 | 4 | 5 | 8 | 5 | 4 | 1 | |||
| F(9,x) | 1 | 4 | 7 | 10 | 10 | 7 | 4 | 1 | ||
| F(10,x) | 1 | 5 | 8 | 16 | 16 | 16 | 8 | 5 | 1 | |
| F(11,x) | 1 | 5 | 10 | 20 | 26 | 26 | 20 | 10 | 5 | 1 |
| F(12,x) | 1 | 6 | 12 | 29 | 38 | 50 | 38 | 29 | 12 | 6 |
| F(13,x) | 1 | 6 | 14 | 35 | 57 | 76 | 76 | 57 | 35 | 14 |
| F(14,x) | 1 | 7 | 16 | 47 | 79 | 126 | 133 | 126 | 79 | 47 |
| F(15,x) | 1 | 7 | 19 | 56 | 111 | 185 | 232 | 232 | 185 | 111 |
| F(16,x) | 1 | 8 | 21 | 72 | 147 | 280 | 375 | 440 | 375 | 280 |
| F(17,x) | 1 | 8 | 24 | 84 | 196 | 392 | 600 | 750 | 750 | 600 |
| F(18,x) | 1 | 9 | 27 | 104 | 252 | 561 | 912 | 1282 | 1387 | 1282 |
| F(19,x) | 1 | 9 | 30 | 120 | 324 | 756 | 1368 | 2052 | 2494 | 2494 |
This is sequence https://oeis.org/A052307.
Full output for n=2..40 in csv format is available here.
The family breakdown is a bit different than above.
FamilyCount( 2, 1): [ 2x 1] FamilyCount( 3, 1): [ 3x 1] FamilyCount( 3, 2): [ 3x 1] FamilyCount( 4, 1): [ 4x 1] FamilyCount( 4, 2): [ 2x 1] [ 4x 1] FamilyCount( 4, 3): [ 4x 1] FamilyCount( 5, 1): [ 5x 1] FamilyCount( 5, 2): [ 5x 2] FamilyCount( 5, 3): [ 5x 2] FamilyCount( 5, 4): [ 5x 1] FamilyCount( 6, 1): [ 6x 1] FamilyCount( 6, 2): [ 3x 1] [ 6x 2] FamilyCount( 6, 3): [ 2x 1] [ 6x 1] [ 12x 1] FamilyCount( 6, 4): [ 3x 1] [ 6x 2] FamilyCount( 6, 5): [ 6x 1] FamilyCount( 7, 1): [ 7x 1] FamilyCount( 7, 2): [ 7x 3] FamilyCount( 7, 3): [ 7x 3] [ 14x 1] FamilyCount( 7, 4): [ 7x 3] [ 14x 1] FamilyCount( 7, 5): [ 7x 3] FamilyCount( 7, 6): [ 7x 1] FamilyCount( 8, 1): [ 8x 1] FamilyCount( 8, 2): [ 4x 1] [ 8x 3] FamilyCount( 8, 3): [ 8x 3] [ 16x 2] FamilyCount( 8, 4): [ 2x 1] [ 4x 1] [ 8x 4] [ 16x 2] FamilyCount( 8, 5): [ 8x 3] [ 16x 2] FamilyCount( 8, 6): [ 4x 1] [ 8x 3] FamilyCount( 8, 7): [ 8x 1] FamilyCount( 9, 1): [ 9x 1] FamilyCount( 9, 2): [ 9x 4] FamilyCount( 9, 3): [ 3x 1] [ 9x 3] [ 18x 3] FamilyCount( 9, 4): [ 9x 6] [ 18x 4] FamilyCount( 9, 5): [ 9x 6] [ 18x 4] FamilyCount( 9, 6): [ 3x 1] [ 9x 3] [ 18x 3] FamilyCount( 9, 7): [ 9x 4] FamilyCount( 9, 8): [ 9x 1] FamilyCount( 10, 1): [ 10x 1] FamilyCount( 10, 2): [ 5x 1] [ 10x 4] FamilyCount( 10, 3): [ 10x 4] [ 20x 4] FamilyCount( 10, 4): [ 5x 2] [ 10x 8] [ 20x 6] FamilyCount( 10, 5): [ 2x 1] [ 10x 5] [ 20x 10] FamilyCount( 10, 6): [ 5x 2] [ 10x 8] [ 20x 6] FamilyCount( 10, 7): [ 10x 4] [ 20x 4] FamilyCount( 10, 8): [ 5x 1] [ 10x 4] FamilyCount( 10, 9): [ 10x 1] FamilyCount( 11, 1): [ 11x 1] FamilyCount( 11, 2): [ 11x 5] FamilyCount( 11, 3): [ 11x 5] [ 22x 5] FamilyCount( 11, 4): [ 11x 10] [ 22x 10] FamilyCount( 11, 5): [ 11x 10] [ 22x 16] FamilyCount( 11, 6): [ 11x 10] [ 22x 16] FamilyCount( 11, 7): [ 11x 10] [ 22x 10] FamilyCount( 11, 8): [ 11x 5] [ 22x 5] FamilyCount( 11, 9): [ 11x 5] FamilyCount( 11, 10): [ 11x 1] FamilyCount( 12, 1): [ 12x 1] FamilyCount( 12, 2): [ 6x 1] [ 12x 5] FamilyCount( 12, 3): [ 4x 1] [ 12x 4] [ 24x 7] FamilyCount( 12, 4): [ 3x 1] [ 6x 2] [ 12x 12] [ 24x 14] FamilyCount( 12, 5): [ 12x 10] [ 24x 28] FamilyCount( 12, 6): [ 2x 1] [ 4x 1] [ 6x 1] [ 12x 18] [ 24x 29] FamilyCount( 12, 7): [ 12x 10] [ 24x 28] FamilyCount( 12, 8): [ 3x 1] [ 6x 2] [ 12x 12] [ 24x 14] FamilyCount( 12, 9): [ 4x 1] [ 12x 4] [ 24x 7] FamilyCount( 12, 10): [ 6x 1] [ 12x 5] FamilyCount( 12, 11): [ 12x 1] FamilyCount( 13, 1): [ 13x 1] FamilyCount( 13, 2): [ 13x 6] FamilyCount( 13, 3): [ 13x 6] [ 26x 8] FamilyCount( 13, 4): [ 13x 15] [ 26x 20] FamilyCount( 13, 5): [ 13x 15] [ 26x 42] FamilyCount( 13, 6): [ 13x 20] [ 26x 56] FamilyCount( 13, 7): [ 13x 20] [ 26x 56] FamilyCount( 13, 8): [ 13x 15] [ 26x 42] FamilyCount( 13, 9): [ 13x 15] [ 26x 20] FamilyCount( 13, 10): [ 13x 6] [ 26x 8] FamilyCount( 13, 11): [ 13x 6] FamilyCount( 13, 12): [ 13x 1] FamilyCount( 14, 1): [ 14x 1] FamilyCount( 14, 2): [ 7x 1] [ 14x 6] FamilyCount( 14, 3): [ 14x 6] [ 28x 10] FamilyCount( 14, 4): [ 7x 3] [ 14x 18] [ 28x 26] FamilyCount( 14, 5): [ 14x 15] [ 28x 64] FamilyCount( 14, 6): [ 7x 3] [ 14x 33] [ 28x 90] FamilyCount( 14, 7): [ 2x 1] [ 14x 19] [ 28x 113] FamilyCount( 14, 8): [ 7x 3] [ 14x 33] [ 28x 90] FamilyCount( 14, 9): [ 14x 15] [ 28x 64] FamilyCount( 14, 10): [ 7x 3] [ 14x 18] [ 28x 26] FamilyCount( 14, 11): [ 14x 6] [ 28x 10] FamilyCount( 14, 12): [ 7x 1] [ 14x 6] FamilyCount( 14, 13): [ 14x 1] FamilyCount( 15, 1): [ 15x 1] FamilyCount( 15, 2): [ 15x 7] FamilyCount( 15, 3): [ 5x 1] [ 15x 6] [ 30x 12] FamilyCount( 15, 4): [ 15x 21] [ 30x 35] FamilyCount( 15, 5): [ 3x 1] [ 15x 20] [ 30x 90] FamilyCount( 15, 6): [ 5x 2] [ 15x 33] [ 30x 150] FamilyCount( 15, 7): [ 15x 35] [ 30x 197] FamilyCount( 15, 8): [ 15x 35] [ 30x 197] FamilyCount( 15, 9): [ 5x 2] [ 15x 33] [ 30x 150] FamilyCount( 15, 10): [ 3x 1] [ 15x 20] [ 30x 90] FamilyCount( 15, 11): [ 15x 21] [ 30x 35] FamilyCount( 15, 12): [ 5x 1] [ 15x 6] [ 30x 12] FamilyCount( 15, 13): [ 15x 7] FamilyCount( 15, 14): [ 15x 1] FamilyCount( 16, 1): [ 16x 1] FamilyCount( 16, 2): [ 8x 1] [ 16x 7] FamilyCount( 16, 3): [ 16x 7] [ 32x 14] FamilyCount( 16, 4): [ 4x 1] [ 8x 3] [ 16x 24] [ 32x 44] FamilyCount( 16, 5): [ 16x 21] [ 32x 126] FamilyCount( 16, 6): [ 8x 3] [ 16x 55] [ 32x 222] FamilyCount( 16, 7): [ 16x 35] [ 32x 340] FamilyCount( 16, 8): [ 2x 1] [ 4x 1] [ 8x 4] [ 16x 66] [ 32x 368] FamilyCount( 16, 9): [ 16x 35] [ 32x 340] FamilyCount( 16, 10): [ 8x 3] [ 16x 55] [ 32x 222] FamilyCount( 16, 11): [ 16x 21] [ 32x 126] FamilyCount( 16, 12): [ 4x 1] [ 8x 3] [ 16x 24] [ 32x 44] FamilyCount( 16, 13): [ 16x 7] [ 32x 14] FamilyCount( 16, 14): [ 8x 1] [ 16x 7] FamilyCount( 16, 15): [ 16x 1] FamilyCount( 17, 1): [ 17x 1] FamilyCount( 17, 2): [ 17x 8] FamilyCount( 17, 3): [ 17x 8] [ 34x 16] FamilyCount( 17, 4): [ 17x 28] [ 34x 56] FamilyCount( 17, 5): [ 17x 28] [ 34x 168] FamilyCount( 17, 6): [ 17x 56] [ 34x 336] FamilyCount( 17, 7): [ 17x 56] [ 34x 544] FamilyCount( 17, 8): [ 17x 70] [ 34x 680] FamilyCount( 17, 9): [ 17x 70] [ 34x 680] FamilyCount( 17, 10): [ 17x 56] [ 34x 544] FamilyCount( 17, 11): [ 17x 56] [ 34x 336] FamilyCount( 17, 12): [ 17x 28] [ 34x 168] FamilyCount( 17, 13): [ 17x 28] [ 34x 56] FamilyCount( 17, 14): [ 17x 8] [ 34x 16] FamilyCount( 17, 15): [ 17x 8] FamilyCount( 17, 16): [ 17x 1] FamilyCount( 18, 1): [ 18x 1] FamilyCount( 18, 2): [ 9x 1] [ 18x 8] FamilyCount( 18, 3): [ 6x 1] [ 18x 7] [ 36x 19] FamilyCount( 18, 4): [ 9x 4] [ 18x 32] [ 36x 68] FamilyCount( 18, 5): [ 18x 28] [ 36x 224] FamilyCount( 18, 6): [ 3x 1] [ 6x 2] [ 9x 3] [ 18x 81] [ 36x 474] FamilyCount( 18, 7): [ 18x 56] [ 36x 856] FamilyCount( 18, 8): [ 9x 6] [ 18x 124] [ 36x 1152] FamilyCount( 18, 9): [ 2x 1] [ 6x 1] [ 12x 1] [ 18x 68] [ 36x 1316] FamilyCount( 18, 10): [ 9x 6] [ 18x 124] [ 36x 1152] FamilyCount( 18, 11): [ 18x 56] [ 36x 856] FamilyCount( 18, 12): [ 3x 1] [ 6x 2] [ 9x 3] [ 18x 81] [ 36x 474] FamilyCount( 18, 13): [ 18x 28] [ 36x 224] FamilyCount( 18, 14): [ 9x 4] [ 18x 32] [ 36x 68] FamilyCount( 18, 15): [ 6x 1] [ 18x 7] [ 36x 19] FamilyCount( 18, 16): [ 9x 1] [ 18x 8] FamilyCount( 18, 17): [ 18x 1] FamilyCount( 19, 1): [ 19x 1] FamilyCount( 19, 2): [ 19x 9] FamilyCount( 19, 3): [ 19x 9] [ 38x 21] FamilyCount( 19, 4): [ 19x 36] [ 38x 84] FamilyCount( 19, 5): [ 19x 36] [ 38x 288] FamilyCount( 19, 6): [ 19x 84] [ 38x 672] FamilyCount( 19, 7): [ 19x 84] [ 38x 1284] FamilyCount( 19, 8): [ 19x 126] [ 38x 1926] FamilyCount( 19, 9): [ 19x 126] [ 38x 2368] FamilyCount( 19, 10): [ 19x 126] [ 38x 2368] FamilyCount( 19, 11): [ 19x 126] [ 38x 1926] FamilyCount( 19, 12): [ 19x 84] [ 38x 1284] FamilyCount( 19, 13): [ 19x 84] [ 38x 672] FamilyCount( 19, 14): [ 19x 36] [ 38x 288] FamilyCount( 19, 15): [ 19x 36] [ 38x 84] FamilyCount( 19, 16): [ 19x 9] [ 38x 21] FamilyCount( 19, 17): [ 19x 9] FamilyCount( 19, 18): [ 19x 1]