A301480 -id:A301480 - cp's OEIS frontend

A300486 Number of relatively prime or monic partitions of n.

1, 2, 3, 4, 7, 8, 15, 18, 28, 35, 56, 64, 101, 120, 168, 210, 297, 348, 490, 583, 776, 946, 1255, 1482, 1952, 2335, 2981, 3581, 4565, 5387, 6842, 8119, 10086, 12013, 14863, 17527, 21637, 25525, 31083, 36695, 44583, 52256, 63261, 74171, 88932, 104303, 124754

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

			The a(6) = 8 relatively prime or monic partitions are (6), (51), (411), (321), (3111), (2211), (21111), (111111). Missing from this list are (42), (33), (222).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A301462, A301467, A301480, A302094, A302698, A302915, A302916, A302917.

Table[Length[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]],{n,20}]

a(n)={(n > 1) + sumdiv(n, d, moebius(d)*numbpart(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n > 1) = 1 + A000837(n) = 1 + Sum_{d|n} mu(d) * A000041(n/d).

A301706 Number of rooted thrice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983

Offset: 1

Gus Wiseman, Mar 25 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

			The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
Array[thrire,30]

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A301764 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 5, 6, 7, 8, 5, 10, 7, 8, 10, 10, 6, 12, 7, 12, 13, 10, 5, 14, 12, 11, 11, 14, 7, 18, 9, 12, 13, 11, 12, 20, 10, 10, 13, 18, 9, 20, 9, 14, 20, 12, 5, 20, 15, 19, 14, 17, 7, 18, 16, 20, 17, 12, 5, 26, 13, 12, 21, 18, 17, 24, 9, 15, 13, 22, 9

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(11) = 8 rooted twice-partitions: (9), (333), (111111111), (4)(4), (22)(22), (1111)(1111), (1)(1)(1)(1)(1), ()()()()()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A007425, A063834, A093637, A127524, A295931, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,DivisorSum[n-1,If[#===1,1,DivisorSigma[0,#-1]]&]],{n,100}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==1, 1, numdiv(d-1)))) \\ Andrew Howroyd, Aug 26 2018

A302094 Number of relatively prime or monic twice-partitions of n.

Original entry on oeis.org

1, 3, 6, 10, 27, 35, 113, 170, 396, 641, 1649, 2318, 5905, 9112, 18678, 32529, 69094, 106210, 227480, 363433, 705210, 1196190, 2325023, 3724233, 7192245, 11915884, 21857887, 36597843, 67406158, 109594872, 201747847, 333400746, 591125465, 987069077, 1743223350

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime). Then a relatively prime or monic twice-partition of n is a choice of a relatively prime or monic partition of each part in a relatively prime or monic partition of n.

			The a(4) = 10 relatively prime or monic twice-partitions:
(4), (31), (211), (1111),
(3)(1), (21)(1), (111)(1),
(2)(1)(1), (11)(1)(1),
(1)(1)(1)(1).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A300486, A301462, A301467, A301480, A302915, A302916, A302917.

ip[n_]:=ip[n]=Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&];
Table[Sum[Times@@Length/@ip/@ptn,{ptn,ip[n]}],{n,10}]

A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 29, 42, 61, 86, 127, 181, 257, 352, 489, 668, 935, 1270, 1730, 2312, 3101, 4112, 5533, 7345, 9742, 12785, 16793, 21821, 28452, 36908, 48108, 62198, 80337, 103081, 132372, 168805, 215247, 273678

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(8) = 18 rooted twice-partitions where the composite rooted partition is strict:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(4)()(), (31)()(), (3)(1)(),
(3)()()(), (21)()()(), (2)(1)()(),
(2)()()()(),
(1)()()()()(),
()()()()()()().

Cf. A002865, A063834, A093637, A127524, A279790, A294788, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Join@@#&]//Length,{n,30}]

A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 63, 82, 109, 140, 183, 233, 298, 376, 479, 598, 753, 938, 1171, 1449, 1797, 2210, 2726, 3342, 4095, 4990, 6088, 7388, 8968, 10843, 13099, 15770, 18975, 22756, 27276, 32603, 38925, 46353, 55158, 65479, 77656, 91904, 108645

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Cf. A002865, A047968, A063834, A093637, A295924, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=(1-nn)/(1-x)+Sum[Product[1/(1-x^(d k+1)),{k,0,nn}],{d,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 26, 32, 36, 42, 52, 59, 66, 79, 93, 108, 125, 141, 162, 192, 222, 248, 285, 331, 375, 430, 492, 555, 632, 719, 816, 929, 1051, 1177, 1327, 1510, 1701, 1908, 2146, 2408, 2705, 3035, 3388, 3792, 4257, 4751, 5284, 5894

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(9) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]

a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ Andrew Howroyd, Aug 26 2018

Terms a(26) and beyond from Andrew Howroyd, Aug 26 2018

A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).

A301753 Number of ways to choose a strict rooted partition of each part in a rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 43, 66, 108, 166, 269, 408, 643, 975, 1517, 2277, 3497, 5223, 7936, 11803, 17736, 26219, 39174, 57594, 85299, 124957, 183987, 268158, 392685, 569987, 830282, 1200843, 1740422, 2507823, 3620550, 5197885, 7472229, 10694865, 15319700

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 16 rooted twice-partitions:
(5), (32), (41),
(2)(2), (3)(1), (4)(), (21)(1), (31)(),
(1)(1)(1), (2)(1)(), (3)()(), (21)()(),
(1)(1)()(), (2)()()(),
(1)()()()(),
()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A270995, A281113, A296119, A301422, A301462, A301467, A301480, A301706, A301751.

nn=50;
ser=x*Product[1/(1-PartitionsQ[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(1/prod(k=1, n-1, 1 - u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} 1/(1 - A000009(n-1) x^n).

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A300486 Number of relatively prime or monic partitions of n.

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A301706 Number of rooted thrice-partitions of n.

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A301595 Number of thrice-partitions of n.

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A301764 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.

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A302094 Number of relatively prime or monic twice-partitions of n.

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A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

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A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

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A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

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