A323429 -id:A323429 - cp's OEIS frontend

A101509 Binomial transform of tau(n) (see A000005).

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022

my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

A323436 Number of plane partitions whose parts are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 7, 2, 4, 2, 3, 1, 7, 2, 5, 2, 2, 1, 8, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 4, 3, 2, 4, 1, 7, 5, 2, 1, 8, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly decreasing.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(120) = 12 plane partitions:
  32111
.
  311   321   3111   3211
  21    11    2      1
.
  31   32   311   321
  21   11   2     1
  1    1    1     1
.
  31   32
  2    1
  1    1
  1    1
.
  3
  2
  1
  1
  1

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A296188, A299968.

Cf. A323300, A323429, A323437, A323438, A323439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,100}]

A089299 Number of square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083

Offset: 0

N. J. A. Sloane, Dec 25 2003

nonn

Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.

			a(7) = 5:
7 41 32 31 22
. 11 11 21 21
a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}
From _Gus Wiseman_, Jan 16 2019: (Start)
The a(10) = 16 square plane partitions:
  [ten]
.
  [32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
  [32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
.
  [211]
  [111]
  [111]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 16 2019 *)

G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006

Corrected and extended by Wouter Meeussen, Dec 30 2003
a(21)-a(25) from John W. Layman, Jan 02 2004
More terms from Franklin T. Adams-Watters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019

A306319 Number of length-rectangular twice-partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 60, 82, 131, 177, 286, 376, 582, 793, 1202, 1610, 2450, 3274, 4906, 6665, 9770, 13274, 19690, 26506, 38596, 53006, 76432, 104189, 150844, 205282, 294304, 404146, 573140, 786169, 1119457, 1527554, 2155953, 2965567, 4163955, 5701816

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is length-rectangular if all parts have the same number of parts.

			The a(5) = 14 length-rectangular twice-partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Dominates A319066 (rectangular partitions of partitions), which dominates A323429 (rectangular plane partitions).

Cf. A000219, A001970, A063834 (twice-partitions), A089299, A271619, A279787 (sum-rectangular twice-partitions), A305551, A306017, A306318 (square case), A323531.

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],SameQ@@Length/@#&],{ptn,IntegerPartitions[n]}]],{n,20}]

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(4) = 13 sequences:
  ()  ((1))  ((2))     ((3))        ((4))
             ((11))    ((21))       ((22))
             ((1)(1))  ((111))      ((31))
                       ((1)(2))     ((211))
                       ((2)(1))     ((1111))
                       ((1)(1)(1))  ((1)(3))
                                    ((2)(2))
                                    ((3)(1))
                                    ((11)(11))
                                    ((1)(1)(2))
                                    ((1)(2)(1))
                                    ((2)(1)(1))
                                    ((1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of set partitions is A038041.
The version for weakly decreasing lengths is A141199, strictly A358836.
For equal sums instead of lengths we have A279787.
The case of twice-partitions is A306319, distinct A358830.
The unordered version is A319066.
The case of plane partitions is A323429.
The case of constant sums also is A358833.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ Andrew Howroyd, Dec 31 2022

G.f.: 1 + Sum_{k>=1} (1/(1 - [y^k]P(x,y)) - 1) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022

A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 12, 16, 22, 27, 36, 44, 57, 72, 89, 110, 139, 170, 210, 261, 318, 390, 478, 581, 705, 860, 1036, 1252, 1511, 1816, 2178, 2618, 3127, 3743, 4471, 5330, 6347, 7564, 8984, 10674, 12669, 15016, 17780, 21050, 24868, 29371, 34655, 40836, 48080

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are strictly decreasing.

			The a(8) = 12 matrices:
  [8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
.
  [7] [6] [5] [3 2]
  [1] [2] [3] [2 1]
.
  [5] [4]
  [2] [3]
  [1] [1]
The a(10) = 22 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [5 2] [4 2] [4 3]
  [1] [2] [3] [4] [2 1] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[Select[IntegerPartitions[#,{k}],UnsameQ@@#&]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323432 Number of semistandard rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 20, 30, 42, 59, 79, 112, 146, 199, 264, 350, 455, 603, 774, 1010, 1297, 1668, 2124, 2724, 3441, 4372, 5513, 6955, 8718, 10960, 13670, 17091, 21264, 26454, 32786, 40667, 50215, 62048, 76435, 94126

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows are weakly decreasing and the columns are strictly decreasing.

			The a(6) = 15 matrices:
  [6] [51] [42] [411] [33] [321] [3111] [222] [2211] [21111] [111111]
.
  [5] [4] [22]
  [1] [2] [11]
.
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323431, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows are weakly increasing and all columns are strictly increasing.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Is this a duplicate of A339887? - R. J. Mathar, Feb 03 2021

			The a(60) = 5 tableaux:
  1123
.
  11   112   113
  23   3     2
.
  11
  2
  3

Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.

Cf. A323300, A323429, A323432, A323436, A323438, A323439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A003293(n).

A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075

Offset: 0

Gus Wiseman, Aug 04 2024

nonn

			The a(0) = 1 through a(4) = 15 ways:
  ()  ((1))  ((2))      ((3))          ((4))
             ((1,1))    ((1,2))        ((1,3))
             ((1),(1))  ((1,1,1))      ((2,2))
                        ((1),(1,1))    ((1,1,2))
                        ((1,1),(1))    ((2),(2))
                        ((1),(1),(1))  ((1,1,1,1))
                                       ((1),(1,2))
                                       ((1,2),(1))
                                       ((1),(1,1,1))
                                       ((1,1),(1,1))
                                       ((1,1,1),(1))
                                       ((1),(1),(1,1))
                                       ((1),(1,1),(1))
                                       ((1,1),(1),(1))
                                       ((1),(1),(1),(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A variation for weakly increasing lengths is A141199.
For identical sums instead of minima we have A279787.
The case of reversed twice-partitions is A306319, distinct A358830.
For maxima instead of minima, or for unreversed partitions, we have A358905.
The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.

Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024

G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - Andrew Howroyd, Dec 29 2024

a(16) onwards from Andrew Howroyd, Dec 29 2024

cp's OEIS Frontend

A101509 Binomial transform of tau(n) (see A000005).

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A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

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A323436 Number of plane partitions whose parts are the prime indices of n.

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A089299 Number of square plane partitions of n.

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A306319 Number of length-rectangular twice-partitions of n.

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A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

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A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

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Mathematica

A323432 Number of semistandard rectangular plane partitions of n.

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