A133121 -id:A133121 - cp's OEIS frontend

A238343 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=n.

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 9, 0, 0, 0, 0, 11, 19, 2, 0, 0, 0, 0, 15, 41, 8, 0, 0, 0, 0, 0, 22, 77, 29, 0, 0, 0, 0, 0, 0, 30, 142, 81, 3, 0, 0, 0, 0, 0, 0, 42, 247, 205, 18, 0, 0, 0, 0, 0, 0, 0, 56, 421, 469, 78, 0, 0, 0, 0, 0, 0, 0, 0, 77, 689, 1013, 264, 5, 0, 0, 0, 0, 0, 0, 0, 0, 101, 1113, 2059, 786, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Counting ascents gives the same triangle.

For n > 0, also the number of compositions of n with k + 1 maximal weakly increasing runs. - Gus Wiseman, Mar 23 2020

			Triangle starts:
00:    1;
01:    1,    0;
02:    2,    0,    0;
03:    3,    1,    0,    0;
04:    5,    3,    0,    0,   0;
05:    7,    9,    0,    0,   0, 0;
06:   11,   19,    2,    0,   0, 0, 0;
07:   15,   41,    8,    0,   0, 0, 0, 0;
08:   22,   77,   29,    0,   0, 0, 0, 0, 0;
09:   30,  142,   81,    3,   0, 0, 0, 0, 0, 0;
10:   42,  247,  205,   18,   0, 0, 0, 0, 0, 0, 0;
11:   56,  421,  469,   78,   0, 0, 0, 0, 0, 0, 0, 0;
12:   77,  689, 1013,  264,   5, 0, 0, 0, 0, 0, 0, 0, 0;
13:  101, 1113, 2059,  786,  37, 0, 0, 0, 0, 0, 0, 0, 0, 0;
14:  135, 1750, 4021, 2097, 189, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
15:  176, 2712, 7558, 5179, 751, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...
From _Gus Wiseman_, Mar 23 2020: (Start)
Row n = 5 counts the following compositions:
  (5)          (3,2)
  (1,4)        (4,1)
  (2,3)        (1,3,1)
  (1,1,3)      (2,1,2)
  (1,2,2)      (2,2,1)
  (1,1,1,2)    (3,1,1)
  (1,1,1,1,1)  (1,1,2,1)
               (1,2,1,1)
               (2,1,1,1)
(End)

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.
T(3n,n) gives A000045(n+1).
T(3n+1,n) = A136376(n+1).
Row sums are A011782.
Compositions by length are A007318.
The version for co-runs or levels is A106356.
The case of partitions (instead of compositions) is A133121.
The version for runs is A238279.
The version without zeros is A238344.
The version for weak ascents is A333213.
Cf. A008284, A045883, A124765, A124766, A332875, A333215.

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, j)*`if`(j (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..20);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*If[jJean-François Alcover, Jan 08 2015, translated from Maple *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],n==0||Length[Split[#,LessEqual]]==k+1&]],{n,0,9},{k,0,n}] (* Gus Wiseman, Mar 23 2020 *)

Sum_{k=0..n} k * T(n,k) = A045883(n-2) for n>=2.

A333213 Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 4, 1, 1, 0, 3, 6, 5, 1, 1, 0, 4, 10, 10, 6, 1, 1, 0, 5, 17, 20, 13, 7, 1, 1, 0, 6, 27, 38, 31, 16, 8, 1, 1, 0, 8, 40, 69, 67, 42, 19, 9, 1, 1, 0, 10, 58, 123, 132, 101, 54, 22, 10, 1, 1

Offset: 0

Gus Wiseman, Mar 14 2020

A composition of n is a finite sequence of positive integers summing to n.
Also the number of compositions of n with k + 1 maximal strictly decreasing subsequences.
Also the number of compositions of n with k adjacent terms that are equal or decreasing (weak descents).

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   1   1
   0   2   4   1   1
   0   3   6   5   1   1
   0   4  10  10   6   1   1
   0   5  17  20  13   7   1   1
   0   6  27  38  31  16   8   1   1
   0   8  40  69  67  42  19   9   1   1
   0  10  58 123 132 101  54  22  10   1   1
   0  12  86 202 262 218 139  67  25  11   1   1
   0  15 121 332 484 467 324 182  81  28  12   1   1
Row n = 6 counts the following compositions:
  (6)    (15)    (114)   (1113)   (11112)  (111111)
  (42)   (24)    (123)   (1122)
  (51)   (33)    (222)   (11121)
  (321)  (132)   (1131)  (11211)
         (141)   (1212)  (12111)
         (213)   (1221)  (21111)
         (231)   (1311)
         (312)   (2112)
         (411)   (2211)
         (2121)  (3111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Compositions by length are A007318.
The case of reversed partitions (instead of compositions) is A008284.
The version counting equal adjacencies is A106356.
The case of partitions (instead of compositions) is A133121.
The version counting unequal adjacencies is A238279.
The strict/strong version is A238343.
Cf. A072704, A107429, A124764, A124769, A329744, A332875, A333230.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#,#1>#2&]]==k&]],{n,0,12},{k,0,n}]

T(n)={my(M=matrix(n+1, n+1)); M[1,1]=x; for(n=1, n, for(k=1, n, M[1+n,1+k] = M[1+n,1+k-1] + x*M[1+n-k, 1+n-k] + (1-x)*M[1+n-k, 1+min(k-1, n-k)])); M[1,1]=1; vector(n+1, i, Vecrev(M[i,i]))}
{ my(A=T(12)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023

A064174 Number of partitions of n with nonnegative rank.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135

Offset: 1

Vladeta Jovovic, Sep 20 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013
Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (22)  (32)   (33)   (43)    (44)
                  (31)  (41)   (42)   (52)    (53)
                        (311)  (51)   (61)    (62)
                               (321)  (322)   (71)
                               (411)  (331)   (332)
                                      (421)   (422)
                                      (511)   (431)
                                      (4111)  (521)
                                              (611)
                                              (4211)
                                              (5111)
Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (331)      (2222)
                    (1111)  (2111)   (2211)    (2221)     (3221)
                            (11111)  (3111)    (3211)     (3311)
                                     (21111)   (4111)     (4211)
                                     (111111)  (22111)    (22211)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Rekha Biswal, bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, Mathoverflow.
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Cf. A063995, A064173 (complement).
Row sums of triangle A105806.
Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.
Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):
map(f, [$1..100]); # Robert Israel, Aug 03 2015

Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],?OddQ]==Count[Last/@Partition[q,2],?OddQ]],{n,16}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

{a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ Seiichi Manyama, May 21 2023

a(n) = (A000041(n) + A047993(n))/2.
a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
A064173(n) + a(n) = A000041(n). - R. J. Mathar, Feb 22 2023
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - Seiichi Manyama, May 21 2023

Mathematica programs modified by Clark Kimberling, Feb 12 2014

A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 3, 3, 7, 6, 11, 12, 15, 21, 25, 31, 43, 49, 58, 79, 89, 108, 135, 165, 190, 232, 279, 328, 387, 461, 536, 650, 743, 870, 1029, 1202, 1381, 1613, 1864, 2163, 2505, 2875, 3292, 3829, 4367, 5001, 5746, 6538, 7462, 8533, 9714, 11008, 12527, 14196

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324519.

			The a(2) = 1 through a(11) = 11 integer partitions:
  (11)  (211)  (221)  (222)  (331)   (611)   (441)   (811)   (551)
               (311)  (411)  (511)   (3221)  (711)   (3322)  (911)
                             (3211)  (4211)  (3222)  (4222)  (3332)
                                             (3321)  (5221)  (4331)
                                             (4221)  (5311)  (4421)
                                             (4311)  (6211)  (5222)
                                             (5211)          (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)

Cf. A003114, A006141, A039900, A046660, A047993, A064174, A090858, A133121.

Cf. A324516, A324518, A324519, A324520.

Table[Length[Select[IntegerPartitions[n],Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A268498 Expansion of Product_{k>=1} ((1 + 2*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 1, 0, 3, -1, 3, 3, 3, 0, 4, 12, 0, 9, -3, 21, 12, 17, -3, 33, 0, 33, 36, 36, 27, 21, 52, 24, 90, 72, 99, 24, 138, 21, 207, 0, 261, 149, 267, 45, 333, 174, 339, 174, 345, 411, 654, 330, 456, 657, 535, 684, 483, 1233, 489, 1353, 882, 1803, 720, 1902, 756

Offset: 0

Vaclav Kotesovec, Feb 06 2016

sign

It appears that this sequence contains only finitely many nonpositive terms, namely at indices {2, 4, 8, 11, 13, 17, 19, 34}. - Gus Wiseman, Jan 23 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A032302, A264686, A268499, A268500.

Cf. A006951, A100471, A133121.

nmax = 100; CoefficientList[Series[Product[(1+2*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(2)^2 + 4*polylog(2, -1/2) = 2.4571173338382709125... .
a(n) = Sum_{k = 0...n} (-1)^k * A133121(n,k). - Gus Wiseman, Jan 23 2019
G.f.: Product_{k>=1} (1 - Sum_{j>=1} (-1)^j * x^(k*j)). - Ilya Gutkovskiy, Nov 06 2019

A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

			The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Christian Sievers, Table of n, a(n) for n = 0..1000

The RHS is counted by A116608, rank statistic A297155.
The LHS is counted by A133121, rank statistic A046660.
For related inequalities see A212165, A212168, A361204.
For subsets instead of partitions see A217615, A385572, A385575.
These partitions are ranked by A385576.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
A355394 counts partitions with neighbors, complement A356236.
Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]

lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ Christian Sievers, Jul 18 2025

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163

Offset: 1

Gus Wiseman, Jul 04 2025

nonn

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.

These are also numbers with the same number of adjacent equal prime indices as adjacent unequal prime indices.

			The prime indices of 2640 are {1,1,1,1,2,3,5}, with 4 distinct parts {1,2,3,5} and 4 maximal anti-runs ((1),(1),(1),(2,3,5)), so 2640 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  28: {1,1,4}
  29: {10}
  31: {11}
  37: {12}
  41: {13}
  43: {14}
  44: {1,1,5}
  45: {2,2,3}
  47: {15}

The LHS is the rank statistic A001221, triangle counted by A116608.
The RHS is the rank statistic A375136, triangle counted by A133121.
These partitions are counted by A385574.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length, ranks A106529.
A356235 counts partitions with a neighborless singleton, ranks A356237.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by maximal anti-runs, for partitions A268193, strict A384905.
A385572 counts subsets with the same number of runs as anti-runs, ranks A385575.
Cf. A044813, A046660, A210034, A297155, A356226, A361205, A384889, A385213.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||PrimeNu[#]==Length[Split[prix[#],UnsameQ]]&]

A001221(a(n)) = A375136(a(n)).

cp's OEIS Frontend

A238343 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=n.

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A333213 Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n.

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A064174 Number of partitions of n with nonnegative rank.

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A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

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A268498 Expansion of Product_{k>=1} ((1 + 2*x^k) / (1 + x^k)).

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A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

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A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

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