A096771 -id:A096771 - cp's OEIS frontend

A325182 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

0, 0, 0, 2, 2, 1, 2, 4, 7, 6, 5, 4, 5, 9, 12, 15, 14, 12, 10, 9, 11, 15, 21, 24, 28, 26, 24, 20, 18, 17, 19, 25, 31, 38, 42, 46, 44, 41, 36, 32, 29, 28, 31, 37, 46, 53, 62, 66, 71, 68, 65, 58, 53, 47, 44, 43, 46, 54, 63, 74, 83, 93, 98, 103, 100, 96, 88, 81

Offset: 0

Gus Wiseman, Apr 08 2019

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

			The a(3) = 2 through a(14) = 12 partitions:
  3    31   311  42    43    44    432   442   533    543    544    554
  111  211       2211  421   422   441   3322  4322   4422   553    5333
                       2221  431   3222  4222  4421   5331   5332   5432
                       3211  2222  3321  4321  33311  33321  5431   5441
                             3221  4221  4411         43311  33322  5531
                             3311  4311                      33331  33332
                             4211                            43321  43322
                                                             44311  43331
                                                             53311  44321
                                                                    44411
                                                                    53321
                                                                    54311

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Wikipedia, Durfee square.

Cf. A006918, A096771, A257990, A263297, A325168.

Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).

Original entry on oeis.org

1, 2, 9, 125, 2401, 161051, 4826809, 410338673, 16983563041, 1801152661463, 420707233300201, 25408476896404831, 6582952005840035281, 925103102315013629321, 73885357344138503765449, 12063348350820368238715343, 3876269050118516845397872321

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The square partition (4,4,4,4) has Heinz number prime(4)^4 = 7^4 = 2401.

Alois P. Heinz, Table of n, a(n) for n = 0..303

After a(0) = 1, same as A062457.
Cf. A002024, A047993, A056239, A096771, A106529, A112798, A115720, A174090, A257990, A263297.
Cf. A088218, A330394.

a:= n-> mul(ithprime(i), i=[n$n]):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 03 2020

Mathematica
```
Table[If[n==0,1,Prime[n]]^n,{n,0,10}]
```

a(n) = A330394(A088218(n)). - Alois P. Heinz, Mar 03 2020

A084835 a(n) = A000094(n+4) - A006918(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 10, 18, 30, 49, 75, 112, 163, 231, 322, 441, 595, 792, 1045, 1361, 1760, 2255, 2871, 3626, 4559, 5691, 7077, 8750, 10780, 13216, 16156, 19662, 23868, 28866, 34828, 41882, 50262, 60138

Offset: 1

Jon Perry, Jul 12 2003

nonn

Also the number of integer partitions of n - 3 with Durfee square of length > 2, i.e., those with at least 3 parts > 2. The Heinz numbers of these partitions are given by A307515. - Gus Wiseman, Apr 12 2019

Cf. A000094, A006918, A096771, A115720, A257990, A307515, A325164, A325192.

A084845 := proc(n)
    A000094(n+4)-A006918(n)
end proc:
seq(A084845(n),n=1..40) ; # R. J. Mathar, May 17 2016

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]>2&]],{n,0,30}] (* Gus Wiseman, Apr 12 2019 *)

A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 7, 1, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 13 2019

			Triangle begins:
  1
  0  0
  0  1  0
  0  0  0  0
  0  0  2  0  0
  0  0  0  1  0  0
  0  0  0  2  1  0  0
  0  0  0  0  2  0  0  0
  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  4  2  0  0  0
  0  0  0  0  0  0  2  5  1  0  0  0
  0  0  0  0  0  0  2  4  7  1  0  0  0
  0  0  0  0  0  0  0  2  6  6  0  0  0  0
  0  0  0  0  0  0  0  2  4  9  7  0  0  0  0
  0  0  0  0  0  0  0  0  2  6 11  5  0  0  0  0
  0  0  0  0  0  0  0  0  2  4 10 14  5  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  6 13 15  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  4 10 19 17  2  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  6 14 22 17  1  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  4 10 21 29 17  1  0  0  0  0
Row n = 16 counts the following partitions:
  (8)         (72)       (64)      (533)    (444)
  (11111111)  (711)      (622)     (542)    (3333)
              (2211111)  (631)     (551)    (4332)
              (3111111)  (6211)    (5222)   (4422)
                         (61111)   (5321)   (4431)
                         (222211)  (5411)
                         (322111)  (32222)
                         (331111)  (33221)
                         (421111)  (33311)
                         (511111)  (42221)
                                   (43211)
                                   (44111)
                                   (52211)
                                   (53111)

Column sums are A000041. Row sums are A325193.

Cf. A051924, A071724, A096771, A115720, A263297, A325178, A325192.

Table[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{n,0,10},{k,0,n}]

A368985 a(n) = sum of the side lengths of the minimum containing squares of all partitions of n.

Original entry on oeis.org

0, 1, 4, 8, 16, 27, 47, 72, 115, 170, 255, 364, 527, 732, 1026, 1401, 1916, 2568, 3451, 4556, 6023, 7859, 10245, 13217, 17041, 21766, 27770, 35173, 44471, 55874, 70092, 87432, 108881, 134951, 166948, 205678, 252951, 309908, 379032, 462046, 562246, 682130

Offset: 0

Andrew Howroyd, Jan 12 2024

nonn

The minimum containing square of a partition has its side length equal to the number of parts or the size of the largest part whichever is greater. a(n) is the sum of the side lengths over all partitions of n.

Alois P. Heinz, Table of n, a(n) for n = 0..750

Cf. A096771, A115995, A182094.

a(n)={my(s=0); if(n, forpart(p=n, s += max(#p, p[#p]))); s}

a(n) = Sum_{k>=1} k*A096771(n,k).

A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 9, 13, 16, 20, 24, 28, 32, 38, 42, 48, 54, 60, 66, 74, 80, 88, 96, 104, 112, 122, 130, 140, 150, 160, 170, 182, 192, 204, 216, 228, 240, 254, 266, 280, 294, 308, 322, 338, 352, 368, 384, 400, 416, 434, 450, 468, 486, 504, 522, 542, 560

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (321)   (322)    (332)     (333)      (433)
         (411)   (331)    (422)     (432)      (442)
         (3111)  (421)    (431)     (441)      (532)
                 (511)    (521)     (522)      (541)
                 (3211)   (611)     (531)      (622)
                 (31111)  (3221)    (621)      (631)
                          (3311)    (711)      (721)
                          (32111)   (3222)     (811)
                          (311111)  (3321)     (3322)
                                    (32211)    (3331)
                                    (33111)    (32221)
                                    (321111)   (33211)
                                    (3111111)  (322111)
                                               (331111)
                                               (3211111)
                                               (31111111)

Column k = 3 of A325227.

Cf. A051924, A096771, A115720, A265283, A325224, A325225, A325229, A325231, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==3&]],{n,30}]

cp's OEIS Frontend

A325182 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

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A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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Maple

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A084835 a(n) = A000094(n+4) - A006918(n).

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Maple

Mathematica

A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

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Mathematica

A368985 a(n) = sum of the side lengths of the minimum containing squares of all partitions of n.

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PARI

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A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

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Mathematica

A307539 Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)...prime(y_k).