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Index to OEIS: Section Par

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Index to OEIS: Section Par


[ Aa | Ab | Al | Am | Ap | Ar | Ba | Be | Bi | Bl | Bo | Br | Ca | Ce | Ch | Cl | Coa | Coi | Com | Con | Cor | Cu | Cy | Da | De | Di | Do | Ea | Ed | El | Eu | Fa | Fe | Fi | Fo | Fu | Ga | Ge | Go | Gra | Gre | Ha | He | Ho | Ia | In | J | K | La | Lc | Li | Lo | Lu | M | Mag | Map | Mat | Me | Mo | Mu | N | Na | Ne | Ni | No | Nu | O | Pac | Par | Pas | Pea | Per | Ph | Poi | Pol | Pos | Pow | Pra | Pri | Pro | Ps | Qua | Que | Ra | Rea | Rel | Res | Ro | Ru | Sa | Se | Si | Sk | So | Sp | Sq | St | Su | Sw | Ta | Te | Th | To | Tra | Tri | Tu | U | V | Wa | We | Wi | X | Y | Z | 1 | 2 | 3 | 4 ]



para-Fibonacci sequences, sequences related to :
para-Fibonacci sequences: A019586*, A035612*
paradoxical sequences, sequences related to :
paradoxical sequences: A053169*, A091967, A031135, A037181
paradoxical sequences: see also diagonal sequences

parasitic numbers: see transposable numbers

parentheses, ways to arrange , sequences related to :
this includes towers of exponents such as 2^2^...^2 (which is A002845)
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, A145545, A145546, A145547, A145548, A145549, A145550, A198683, A049006, A077589, A077590
parentheses, ways to arrange: see also Catalan numbers

parenthesized in 2 ways: A006895

PARI , sequences related to :
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*
PARI: see also Dirichlet series

parity sequence: A010060 partial products are primes: A036013, A046966, A046972, A051957, A073673, A073674, A083769, A083770, A083771, A084401, A084402, A084724. A087338

partial sums , sequences related to :
partial sum of (n mod m): A130481 (m=3), A130482, A130483, A130484 (m=6).
partial sums of powers (a.k.a. q-integers): (q^n-1)/(q-1) for q=2,...,50:
A000225, A003462, A002450, A003463 (q=5), A003464, A023000, A023001, A002452, A002275 (q=10), A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108 (q=20), A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733 (q=30), A218734, A218735, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743 (q=40), A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752 (q=50), A094028 (q=100).
negative q = -2, ..., -15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).
See also the index entry for q-integers.
See also the index entry for powers.
partial sum of other sequences: A064831 (areas of Fibonacci rectangles)
partial sum of sequences related to binomial coefficients:
A014137 (Catalan numbers), A014138 (Catalan numbers starting 1, 2, 5...), A064831 (areas of Fibonacci rectangles), A008949 (binomial coefficients)
partial sum of sequences related to digits: A102685 (nonzero digits in n)

partially ordered sets: see posets
partially ordered sets: see also Lattices

partition functions for lattices, sequences related to :
partition functions for lattices: A002890, A002891, A001393, A002892, A001407, A001406

partitioning a polygon: see Index entry for dissections, sequences related to
partitioning a triangle: see Index entry for dissections, sequences related to

partitions , sequences related to :
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, bipartite: see "partitions, into pairs"
partitions, circular: A037306, A047996, A008965. A000031, A215251
partitions, complete: A126796
partitions, congruences for: A213260, A071734, A213261, A071746, A213256, A076394
partitions, graphical: A000569*, A004250*, A004251*, A029889*, A007721* (connected graphs)
partitions, graphical: see also graphical partitions
partitions, graphical: see also A007722, A029890, A029891, A029892, A029893, A029894, A029895
partitions, into distinct parts:
"partitions of n into distinct parts >= k" and "partitions of n into distinct parts, the least being k-1" come in pairs of closely related sequences: A025147, A096765 (k=2); A025148, A096749 (k=3); A025149, A026824 (k=4); A025150, A026825 (k=5); A025151, A026826 (k=6); A025152, A026827 (k=7); A025153, A026828 (k=8); A025154, A026829 (k=9); A025155, A026830 (k=10); A096740, A026831 (k=11)
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, A054225, A054242, 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 (also compositions into relatively prime parts): A282749, A051424; A282750, A000837; A282748, A101268; A101391, A000740
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, notes on (01): 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, 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 5n: see separate page Partitions of 5n
partitions, of points on a circle, A001005
partitions, of unity, A002966, A006585
partitions, order-consecutive, A007052
partitions, partition numbers, prime: A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170, A114171
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: see also expansions of product_{k >= 1} (1-x^k)^m
partitions: see also under compositions

[ Aa | Ab | Al | Am | Ap | Ar | Ba | Be | Bi | Bl | Bo | Br | Ca | Ce | Ch | Cl | Coa | Coi | Com | Con | Cor | Cu | Cy | Da | De | Di | Do | Ea | Ed | El | Eu | Fa | Fe | Fi | Fo | Fu | Ga | Ge | Go | Gra | Gre | Ha | He | Ho | Ia | In | J | K | La | Lc | Li | Lo | Lu | M | Mag | Map | Mat | Me | Mo | Mu | N | Na | Ne | Ni | No | Nu | O | Pac | Par | Pas | Pea | Per | Ph | Poi | Pol | Pos | Pow | Pra | Pri | Pro | Ps | Qua | Que | Ra | Rea | Rel | Res | Ro | Ru | Sa | Se | Si | Sk | So | Sp | Sq | St | Su | Sw | Ta | Te | Th | To | Tra | Tri | Tu | U | V | Wa | We | Wi | X | Y | Z | 1 | 2 | 3 | 4 ]


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