A064173 -id:A064173 - cp's OEIS frontend

A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 7, 10, 10, 13, 14, 19, 21, 27, 31, 40, 45, 55, 64, 79, 91, 111, 127, 154, 177, 211, 243, 290, 333, 394, 455, 538, 618, 726, 834, 977, 1121, 1304, 1495, 1738, 1989, 2302, 2633, 3041, 3473, 3999, 4562, 5241

Offset: 1

Clark Kimberling, Feb 13 2014

Also, the number of partitions of n such that (greatest part) = 3*(number of parts).

			a(15) = 4 counts these partitions: [12,1,1,1], [9,5,1], [9,4,2], [9,3,3].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Column 3 of A350879.

Cf. A064173, A237753, A350892, A350894.

z = 50; Table[Count[IntegerPartitions[n], p_ /; Max[p] = = 3 Length[p]], {n, z}]
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^(4*k-1) * Product[(1 - x^(3*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/4 + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 15 2024 *)
nmax = 100; p = x^2; s = x^2; Do[p = Normal[Series[p*x^4*(1 - x^(4*k - 1))*(1 - x^(4*k))*(1 - x^(4*k + 1))*(1 - x^(4*k + 2))/((1 - x^(3*k + 2))*(1 - x^(3*k + 1))*(1 - x^(3*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/4 + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(4*k-1)*prod(j=1, k-1, (1-x^(3*k+j-1))/(1-x^j))))) \\ Seiichi Manyama, Jan 24 2022

G.f.: Sum_{k>=1} x^(4*k-1) * Product_{j=1..k-1} (1-x^(3*k+j-1))/(1-x^j). - Seiichi Manyama, Jan 24 2022

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(5/2)). - Vaclav Kotesovec, Oct 17 2024

A325342 Number of partitions p of n such that min(p) < (number of parts of p) <= max(p).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 24, 33, 46, 61, 82, 108, 142, 184, 239, 305, 391, 495, 626, 786, 985, 1226, 1524, 1884, 2323, 2853, 3497, 4268, 5200, 6314, 7650, 9243, 11146, 13403, 16090, 19268, 23032, 27473, 32716, 38878, 46130, 54633, 64603, 76264, 89899

Offset: 1

Clark Kimberling, Apr 21 2019

nonn

			a(7) counts these 6 partitions: {6,1}, {5,1,1}, {4,2,1}, {4,1,1,1}, {3,3,1}, {3,2,2}.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A000041, A003114, A064173, A325340, A325341, A325343.

Table[Count[IntegerPartitions[n], q_ /; Min[q] < Length[q] <= Max[q]], {n, 60}]

a(n) = A000041(n) - A003114(n) - A064173(n). - Jason Yuen, Dec 17 2024

A325343 Number of partitions p of n such that min(p) <= (number of parts of p) <= max(p).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 50, 65, 87, 114, 149, 192, 249, 316, 404, 510, 643, 805, 1008, 1251, 1553, 1917, 2361, 2895, 3546, 4322, 5262, 6383, 7728, 9330, 11245, 13512, 16213, 19405, 23186, 27643, 32907, 39089, 46366, 54894, 64893, 76584, 90256

Offset: 1

Clark Kimberling, Apr 21 2019

nonn

			a(7) counts these 7 partitions: {6,1}, {5,2}, {5,1,1}, {4,2,1}, {4,1,1,1}, {3,3,1}, {3,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A003106, A064173, A325340, A325341, A325342.

Table[Count[IntegerPartitions[n], q_ /; Min[q] <= Length[q] <= Max[q]], {n, 60}]

a(n) = A000041(n) - A003106(n) - A064173(n) for n > 0. - Jason Yuen, Dec 15 2024

A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

Original entry on oeis.org

0, 2, 3, 4, 6, 10, 14, 21, 28, 40, 53, 74, 97, 131, 171, 225, 290, 377, 480, 616, 779, 987, 1238, 1556, 1935, 2411, 2981, 3685, 4527, 5562, 6793, 8295, 10081, 12241, 14805, 17890, 21538, 25906, 31062, 37201, 44429, 53004, 63070, 74964, 88898, 105297

Offset: 1

Clark Kimberling, Apr 02 2014

Let # denote "number of" and c(p) = conjugate of partitionp.  Then
A240057(n) = # p such that min(p) not = max(c(p));
A039899(n) = # p such that min(p) < max(c(p));
A039900(n) = # p such that min(p) <= max(c(p));
A006141(n) = # p such that min(p) = max(c(p));
A003114(n) = # p such that min(p) > max(c(p));
A003016(n) = # p such that min(p) >= max(c(p));
A064173(n) = # p such that max(p) < max(c(p));
A064174(n) = # p such that max(p) <= max(c(p));
A047993(n) = # p such that max(p) = max(c(p)).
See A240178 for related sequences. - Clark Kimberling, Apr 11 2014

			a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003106, A003114, A006141, A039899, A039900.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
    end:
a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 03 2014

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A006141(n) = A000041(n) for n > 0.

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

A098125 Number of compositions of n where the largest part is less than the number of parts.

Original entry on oeis.org

0, 1, 1, 4, 8, 17, 38, 83, 174, 367, 771, 1606, 3324, 6849, 14054, 28743, 58605, 119161, 241717, 489345, 988945, 1995604, 4021710, 8095815, 16281400, 32716231, 65694106, 131833462, 264423116, 530128036, 1062424867, 2128513095

Offset: 1

Vladeta Jovovic, Sep 25 2004

			a(5)=8 because we have 1112, 1121, 1211, 2111, 122, 212, 221 and 11111.

Cf. A077229, A047993, A064173, A064174.

G:=sum(((x^k-x)/(x-1))^k,k=0..45): Gser:=series(G,x=0,40): seq(coeff(Gser,x^n),n=1..36); # Emeric Deutsch, Apr 16 2005

G.f.: Sum_{k>=0} ((x^k-x)/(x-1))^k.

More terms from Emeric Deutsch, Apr 16 2005

A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts.

Original entry on oeis.org

0, 2, 2, 4, 6, 10, 12, 20, 26, 38, 50, 70, 90, 124, 160, 212, 272, 356, 450, 582, 732, 932, 1166, 1470, 1824, 2280, 2814, 3486, 4280, 5268, 6428, 7864, 9552, 11614, 14044, 16990, 20450, 24626, 29524, 35392, 42272, 50472, 60060, 71444, 84734, 100432, 118736

Offset: 1

Omar E. Pol, Feb 18 2014

nonn

Number of partitions of n whose rank is not 0.

			For n = 5 we have:
-------------------------------------------------------
Partitions    Largest    Number    Dyson's
of 5           part     of parts    rank       Type
-------------------------------------------------------
5                5    -    1    =    4      unbalanced
4+1              4    -    2    =    2      unbalanced
3+2              3    -    2    =    1      unbalanced
3+1+1            3    -    3    =    0      balanced
2+2+1            2    -    3    =   -1      unbalanced
2+1+1+1          2    -    4    =   -2      unbalanced
1+1+1+1+1        1    -    5    =   -4      unbalanced
-------------------------------------------------------
There are 6 partitions whose rank is not 0, so a(5) = 6.

Cf. A000041, A047993, A064173, A209616.

P = PartitionsP;
a[n_] := P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}];
a /@ Range[46] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

a(n) = A000041(n) - A047993(n) = 2*A064173(n).

A340929 Heinz numbers of integer partitions of odd negative rank.

Original entry on oeis.org

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
       4: (1,1)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101707.
The positive version is A101707 (A340604).
The even version is A101708 (A340930).
The not necessarily odd version is A064173 (A340788).
A001222 counts prime factors.
A027193 counts partitions of odd length (A026424).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.

A340930 Heinz numbers of integer partitions of even negative rank.

Original entry on oeis.org

8, 24, 32, 36, 54, 80, 81, 96, 120, 128, 144, 180, 200, 216, 224, 270, 300, 320, 324, 336, 384, 405, 450, 480, 486, 500, 504, 512, 560, 576, 675, 704, 720, 729, 750, 756, 784, 800, 840, 864, 896, 1056, 1080, 1125, 1134, 1176, 1200, 1250, 1260, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
       8: (1,1,1)             270: (3,2,2,2,1)
      24: (2,1,1,1)           300: (3,3,2,1,1)
      32: (1,1,1,1,1)         320: (3,1,1,1,1,1,1)
      36: (2,2,1,1)           324: (2,2,2,2,1,1)
      54: (2,2,2,1)           336: (4,2,1,1,1,1)
      80: (3,1,1,1,1)         384: (2,1,1,1,1,1,1,1)
      81: (2,2,2,2)           405: (3,2,2,2,2)
      96: (2,1,1,1,1,1)       450: (3,3,2,2,1)
     120: (3,2,1,1,1)         480: (3,2,1,1,1,1,1)
     128: (1,1,1,1,1,1,1)     486: (2,2,2,2,2,1)
     144: (2,2,1,1,1,1)       500: (3,3,3,1,1)
     180: (3,2,2,1,1)         504: (4,2,2,1,1,1)
     200: (3,3,1,1,1)         512: (1,1,1,1,1,1,1,1,1)
     216: (2,2,2,1,1,1)       560: (4,3,1,1,1,1)
     224: (4,1,1,1,1,1)       576: (2,2,1,1,1,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101708.
The positive version is (A340605).
The odd version is A101707 (A340929).
The not necessarily even version is A064173 (A340788).
A001222 counts prime factors.
A027187 counts partitions of even length.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058696 counts partitions of even numbers.
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A006141, A039900, A117193, A117409, A316413, A324517, A325134, A326845, A340604, A340787, A340828, A340830.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],EvenQ[rk[#]]&&rk[#]<0&]

A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 120, 154, 201, 256, 330, 415, 529, 662, 833, 1035, 1293, 1595, 1976, 2425, 2982, 3640, 4449, 5401, 6565, 7935, 9592, 11543, 13891, 16645, 19943, 23808, 28408, 33792, 40172, 47619, 56413, 66661, 78708, 92724, 109149, 128213, 150486, 176293

Offset: 1

Seiichi Manyama, May 22 2023

nonn

			a(6) = 2 counts these partitions: 6, 5+1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), this sequence (r=3), A363231 (r=4).

Cf. A000041, A101200, A115067.

a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+5)/2));

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+5)/2).
a(n) = p(n-4) - p(n-11) + p(n-21) - ... + (-1)^(k-1) * p(n-k*(3*k+5)/2) + ..., where p() is A000041().
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 31*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

cp's OEIS Frontend

A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

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A325342 Number of partitions p of n such that min(p) < (number of parts of p) <= max(p).

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A325343 Number of partitions p of n such that min(p) <= (number of parts of p) <= max(p).

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A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

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A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A098125 Number of compositions of n where the largest part is less than the number of parts.

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A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts.

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A340929 Heinz numbers of integer partitions of odd negative rank.

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