A096401 -id:A096401 - cp's OEIS frontend

A237976 Number of strict partitions of n such that (least part) < number of parts.

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 14, 17, 21, 25, 31, 37, 45, 54, 64, 76, 90, 106, 124, 146, 170, 198, 230, 267, 308, 357, 410, 472, 542, 621, 709, 811, 923, 1051, 1194, 1355, 1534, 1738, 1962, 2215, 2497, 2812, 3161, 3553, 3986, 4469, 5005, 5600, 6258

Offset: 0

Clark Kimberling, Feb 18 2014

			a(8) = 3 counts these partitions:  71, 521, 431.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A237977, A096401, A237979, A025157, A039899.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A025157(n). - Vaclav Kotesovec, Jan 18 2022

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

A237977 Number of strict partitions of n such that (least part) <= number of parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 29, 36, 42, 51, 60, 72, 84, 100, 117, 137, 160, 187, 216, 251, 290, 334, 385, 442, 507, 581, 664, 757, 864, 982, 1116, 1266, 1435, 1622, 1835, 2069, 2333, 2626, 2954, 3316, 3724, 4172, 4673, 5227, 5844

Offset: 0

Clark Kimberling, Feb 18 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A237976, A096401, A237979, A025157, A039900.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A237979(n). - Vaclav Kotesovec, Jan 18 2022

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

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}]

A339446 Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 62, 62, 92, 116, 146, 170, 224, 248, 302, 350, 404, 452, 530, 698, 776, 968, 1166, 1478, 1700, 2132, 2474, 3050, 3512, 4208, 4814, 5750, 6476, 7556, 8522, 10562, 11672, 13952, 16022, 19286, 22316, 26540

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(12) = 8 because we have [10, 2], [2, 10], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 5, 4] and [3, 4, 5].

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Index entries for sequences related to compositions

Cf. A006141, A032020, A096401, A098133.

nmax = 60; CoefficientList[Series[Sum[k! x^(k (3 k - 1)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} k! * x^(k*(3*k - 1)/2) / Product_{j=1..k-1} (1 - x^j).

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.

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A237976 Number of strict partitions of n such that (least part) < number of parts.

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A237977 Number of strict partitions of n such that (least part) <= number of parts.

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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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A339446 Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.

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A340929 Heinz numbers of integer partitions of odd negative rank.

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