para-Fibonacci sequences: A019586 *, A035612 *
paradoxical sequences: A031135 *, A037181 , A053169
paradoxical sequences: see also diagonal sequences
parentheses, ways to arrange , sequences related to (start):
parentheses, ways to arrange: (1) A000081 * A000108 * A001003 * A001190 * A001699 * A047929 A054026 A057546 A061855 A071153 A075729 A078623
parentheses, ways to arrange: (2) A079216 A079217 A000311 A001147 A002845 A003006 A003007 A003008 A003018 A003019
parenthesized in 2 ways: A006895
PARI code for printing a square array or table by antidiagonals: A025581 *, A002262 *, A004736 *, A002260 *, A004070 *
PARI code for printing a triangle by rows: A003056 *, A002024 *, A003057 *, A055086 *, A073188 *, A000194 *
PARI code for sequences obtained by concatenating strings: A005713 *
PARI code for sequences obtained by repeated substitutions: A005614 *
PARI code for set of digits of n in base k: A000695 *
parity sequence: A010060
partially ordered sets: see posets
partially ordered sets: see also Lattices
partition function for lattices: A002890 , A002891 , A001393 , A002892 , A001407 , A001406
partition numbers, prime: A046063 , A114165 , A111389 , A111045 , A114166 , A111036 , A114167 , A114168 , A114169 , A114170 , A114171 .
partitions , sequences related to (start):
partitions, A000041 *
Partitions, A002300 , A007209 , A002099 , A001144 , A002098 , A000065 , A002622 , A002040 , A007312 , A002039 , A002164 , A006628
partitions, average number of parts: see A006128
partitions, binary: A000123 *, A018819
partitions, graphical: A000569 *, A004250 *, A004251 *, A029889 *, A007721 * (connected graphs)
partitions, graphical: see also A007722 , A029890 , A029891 , A029892 , A029893 , A029894 , A029895
partitions, graphical: see also graphical partitions
partitions, into distinct parts: A000009 *, A000700 (distinct odd parts)
partitions, into distinct primes: A000586 *
partitions, into even number of parts: A027187
partitions, into Fibonacci numbers: see Fibonacci numbers, number of ways to write n as a sum of
Partitions, into non-integral powers, A000135 , A000148 , A000158 , A000160 , A000234 , A000263 , A000298 , A000327 , A000333 , A000339 , A000345 , A000347 , A000397
partitions, into odd number of parts: A027193
partitions, into odd parts: A000009
Partitions, into pairs, A006199 , A006198 , A006200 , A090806
partitions, into parts 5k+-1: A003114 *
partitions, into parts 5k+-2: A003106 *
Partitions, into parts of m kinds, A000070 , A000097 , A000098 , A000710 , A000712 , A000713 , A000714 , A000715 , A000711 , A000716
Partitions, into powers, A003108 , A005706 , A005705 , A005704 , A002572
Partitions, into prime parts, A000586 , A007359 , A002100 , A007360 , A000607 *, A002095 , A000726
partitions, into primes: A000607 *, A000586 (distinct primes)
partitions, into relatively prime parts: A051424 *
partitions, into triangular numbers: A007294
partitions, m-ary: A000123 , A018819 , A005704 , A005705 , A005706
Partitions, maximal, A002569
Partitions, mixed, A002096
Partitions, multi-dimensional, A000334 , A000390 , A000416 , A000427 , A002721
Partitions, multi-line, A003292 , A000990 , A000991 , A002799 , A001452
partitions, non-squashing: A000123 , A018819 , A088567 , A088575 , A088585 , A089300 , A089292
partitions, number of parts in all: A006128
partitions, numbers n such that P(k*n) is prime, where P(n) is the number of partitions of n: A046063 , A114165 , A111389 , A111045 , A114166 , A111036 , A114167 , A114168 , A114169 , A114170 , A113499 , A115214
partitions, odd: A000009
partitions, of a polygon: A002058 , A002059 , A002060
partitions, of a polygon: see also dissections
partitions, of n into 4 squares: A002635 *
partitions, of n into 4th powers: A046042 *
Partitions, of points on a circle, A001005
Partitions, of unity, A002966 , A006585
Partitions, order-consecutive, A007052
partitions, perfect: A002033
partitions, planar: A000219 *, A001522 , A001523 , A001524 , A089300 , A089299 , A089292
Partitions, planar:: A000784 , A005987 , A000786 , A003293 , A000785 , A005986 , A005157 , A006366 , A002659 , A002660 , A002791 , A002800
partitions, protruded: A005403 , A005404 , A005405 , A005406 , A005407 , A005116
Partitions, refinements of, A002846
partitions, restricted (1):: A002637 , A002635 , A002471 , A002636 , A007690 , A001156 , A007294 , A003105 , A003106 , A003114
partitions, restricted (2):: A002865 , A001399 , A006950 , A001972 , A007279 , A001971 , A001400 , A001401 , A001402 , A002573
partitions, restricted (3):: A002574 , A002843 , A005895 , A006827 , A007511 , A005896 , A001976 , A001975 , A002219 , A001978
partitions, restricted (4):: A006477 , A001977 , A001980 , A001979 , A002220 , A001982 , A001981 , A002221 , A002222
Partitions, rotatable, A002722 , A002723
partitions, solid (1): A000293 * A000294 A002835 A002836 A005980 A037452 A080207 A002043 A002936 A002974 A002044 A002045
partitions, solid (2): A082535
partitions, square: A008763 , A089299
partitions, total number of parts: A006128
partitions, total: A000311 * (labeled), A000669 * (labeled)
partitions, total: see also total orders
partitions, triangle of number of partitions of n in which greatest part is k: A008284 *
partitions, triangle of number of partitions of n into k parts: A008284 *
partitions, wide: A070830
partitions, | notes on (01): (Courtesy of Bob Proctor) When considering partitions of n (initially labeled) objects, we may:
partitions, | notes on (02): (1) Allow the "blocks" to be empty - so more generally refer to "pieces".
partitions, | notes on (03): (2) Order the pieces - so consider "sequences" of pieces instead of "collections".
partitions, | notes on (04): (3) Order the elements within the pieces - so consider "lists" instead of "sets".
partitions, | notes on (05): (4) Erase the labels on the objects - this produces partitions or compositions of integers.
partitions, | notes on (06): With these considerations in mind, we define 6 rows of a table. The columns are defined by formulating various conditions on how many objects can be in the pieces. The six rows are:
partitions, | notes on (07): Row A: Sequences of lists of labeled elements (e.g. books on shelves)
partitions, | notes on (08): Row B: Sequences of sets of labeled elements (i.e. ordered partitions)
partitions, | notes on (08): Row C: Sequences of multisets on one color of marble (i.e. compositions)
partitions, | notes on (09): Row D: Collections of lists of labeled elements (e.g. stacks of books)
partitions, | notes on (10): Row E: Collections of sets of labeled elements (i.e. set partitions)
partitions, | notes on (11): Row F: Collections of multisets on one color of marble (i.e. integer partitions)
partitions, | notes on (12): In the columns, m is the number of marbles and b is the number of bins.
partitions, | notes on (13): Column 1: m elements. Each block has at least 1 element (and number of blocks varies)
partitions, | notes on (14): Column 2: m elements. Each block has at least 2 elements (and number of blocks varies)
partitions, | notes on (15): Column 3: m elements. Each block has 1 or 2 elements (and number of blocks varies)
partitions, | notes on (16): Column 4: b blocks. Each block has exactly 2 elements (and there are 2b elements)
partitions, | notes on (17): Column 5: b pieces. Each piece has 0 or 1 elements (and number of elements varies)
partitions, | notes on (18): Column 6: b pieces. Each piece has 0, 1, or 2 elements (and number of elements varies)
partitions, | notes on (19): Column 7: b blocks. Each block has 1 or 2 elements (and number of elements varies)
partitions, | notes on (20): OEIS # Col 1 Col 2 Col 3 Col 4 Col 4 Col 6 Col 7
partitions, | notes on (21): Row A A002866 A052554 A005442 A010050 A000522 A082765 A099022
partitions, | notes on (22): Row B A000670 A032032 A080599 A000680 A000522 A003011 A105749
partitions, | notes on (23): Row C A011782 A000045 A000045 A000012 A000079 A000244 A000079
partitions, | notes on (24): Row D A000262 A052845 A047974 A001813 A000027 A105747 A001517
partitions, | notes on (25): Row E A000110 A000296 A000085 A001147 A000027 A105748 A001515
partitions, | notes on (26): Row F A000041 A002865 A008619 A000012 A000027 A000217 A000027
partitions, | notes on (27): Reference: R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions! arXiv math.CO.0606404.
partitions: see also expansions of product_{k >= 1} (1-x^k)^m
partitions: see also under compositions