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You are here: Index location The Database of Number Correlations location Partition Theory

Partition Theory

Partition theory is a fundamental area of number theory. It is concerned with the number of ways that a whole number can be partitioned into whole number parts.

5 for example can be partitioned in 7 ways thus:
11111, 2111, 221, 311, 32, 41, 5.
The permutations of these 7 partitions add up to 16 thus:
11111 =1 permutation
2111 =4 permutations
221 =3 permutations
311 =3 permutations
32 =2 permutations
41 =2 permutations
5 =1 permutation


A partition is a way of writing an integer n as a sum of positive integers where the order of the parts is not significant.
Number Partitions Order counts
0 1 1
1 1 1
2 2 2
3 3 4
4 5 8
5 7 16
6 11 32
7 15 64
8 22 128
9 30 256
10 42 512
11 56 1024
12 77 2048
13 101 4096
14 135 8192
15 176 16384
16 231 -
17 297 -
18 385 -
19 490 -
20 627 -
21 792 -
22 1002 -
http://home.att.net/~numericana/data/partition.htm Partition Numbers for 0 to 4096 (site down?)

5, 7, 11 Congruences

Starting with 4, the number of partitions for every 5th integer is a multiple of 5-eg the number of partitions for 9 is 30 and for 14 is 135.

Starting with 5, every 7th integer is a multiple of 7.

Starting with 6, every 11th integer is a multiple of 11.

Other Relationships

Number of partitions of n into an even number of distinct parts equals the number of partitions of n into an odd number of distinct parts-except when n is a Pentagonal Number.(Franklin?)

Number of partitions of n into parts not divisible by 3>than the number of partitions of n into parts none of which occurs more than twice.

The number of odd partitions of n=the number of distinct partitions of n.

The total number of 1s that occur among all unordered partitions of a positive integer is equal to the sum of the numbers of distinct members of those partitions.

SEQUENCE: 0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637 ...
The total number of parts in all partitions of n. Also sum of all large parts of all partitions of n.

SEQUENCE: 0, 0, 1, 3, 7, 13, 24, 39, 64, 98, 150, 219, 322, 455, 645, 892, 1232, 1668, 2259, 3008, 4003, 5260, 6897, 8951, 11599, 14893, 19086, 24284, 30827, 38888, 48959, 61293, 76578, 95223, 118152, 145993, 180037, 221175, 271186, 331402, 404208, 491521...
The number of "+" signs needed to write the partitions of n.

Number of partitions of n into distinct parts >= 2....number of partitions of n into odd parts.
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4097, 4582, 5120, 5718
http://www.research.att.com/~njas/sequences/?q=distinct+parts&sort=0&fmt=0&language=english&go=Search



You are here: Index location The Database of Number Correlations location Partition Theory




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Contents

1) Introduction
2) How to read This book
3) The Virtues
4) Examples of virtuous opposites
5) *Exercise one: Overview
6) Summary of Exercise one: Overview
7) Beliefs
8) Two attitudes towards Truth
9) *Exercise two: Letting go of false beliefs
10) Summary of Exercise Two: Letting go of false beliefs
11) More About the Virtues
12) *Exercise Three: Awakening the Virtues
13) The Virtues are Interconnected
14) *Exercise Four: Reuniting Virtues
15) Every Virtue is Useful in Awakening the Other Virtues
16) Examples of Individual Pathways
17) *Exercise Five: Unifying Your Virtuous Matrix
18) Vices: Yours
19) *Exercise Six: Dealing with Your Own Faults
20) Summary of Exercise Six: Dealing with Your Own Faults
21) Vices: Other Peoples
22) *Exercise Seven: Accepting Others Despite Their Faults
23) A Deeper Understanding of the Matrix
24) Compound Virtues
25) Gaps in the Matrix
26) Beyond the Elemental
27) The Unknown Virtue
28) *Exercise Eight: Reintegrating the Unknown Virtue
29) Utilising Semiconciousness
30) Mental Association
31) *Exercise Nine: Positive Association
32) Social an Universal Association
33) *Exercise Ten: Enhancing Your Surroundings
34) Anchoring Positive Association
35) *Exercise Eleven: Anchoring Positive Association
36) Summary of Exercise Eleven: Anchoring Positive Association
37) Exercise Twelve: Deepening Positive Association
38) Individual Areas
39) *Exercise Thirteen: Individual Areas
40) Independence!
41) *Exercise Fourteen: Reclaiming Your Power
42) Summary of Exercise Fourteen: Reclaiming Your Power
43) The Darkness Within
44) *Exercise Fifteen: Embracing the Darkness
45) Summary of Exercise Fifteen: Embracing the Darkness
46) Improving Your Practice
47) *Exercise Sixteen: Boosting Practice
48) Final Word
Inner Medicine by James Barton