cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160
Offset: 1

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Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions satisfying (maximum) = 2*(mean).
These are partitions whose diagram has the same size as its complement (see example).

Examples

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

Crossrefs

For minimum instead of mean we have A118096.
For length instead of mean we have A237753.
For median instead of mean we have A361849, ranks A361856.
This is the equal case of A361851, unequal case A361852.
The strict case is A361854.
These partitions have ranks A361855.
This is the equal case of A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#==2n&]],{n,30}]

A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 2, 2, 2, 0, 2, 1, 4, 0, 2, 1, 2, 0, 2, 2, 2, 2, 2, 0, 4, 0, 0, 2, 1, 4, 1, 2, 0, 6, 0, 2, 2, 1, 0, 2, 0, 2, 3, 2, 0, 6, 0, 2, 4, 4, 0, 2, 0, 2, 2, 0, 0, 2, 1, 4, 0, 6, 0, 2, 4, 5, 0, 6, 0, 4, 2, 0, 0, 4, 1, 0, 0, 2, 0, 2, 2, 4, 0, 2, 6, 5, 0, 6, 0, 8
Offset: 1

Views

Author

Emeric Deutsch, Feb 12 2016

Keywords

Comments

The complement of a partition p[1] >= p[2] >=...>= p[k] is p[1]-p[2], p[1]-p[3], ..., p[1]-p[k]. Its Ferrers board emerges naturally from the Ferrers board of the given partition. The weight of a partition of n is n.
Sum of entries in row n is A000041(n) (the partition numbers).
Apparently, number of entries in row n is A033638(n-1) = 1 + floor((n-1)^2/4).
T(n,0) = A000005(n) = number of divisors of n.
T(n,1) = A070824(n+1).
Sum(k*T(n,k),k>0) = A188814(n).

Examples

			Row 4 is 3,0,2; indeed, the complements of [4], [3,1], [2,2], [2,1,1], [1,1,1,1] are: empty, [2], empty, [1,1], empty; their weights are 0, 2, 0, 2, 0, respectively.
From _Gus Wiseman_, Sep 24 2019: (Start)
Triangle begins:
  1
  2
  2 1
  3 0 2
  2 2 0 2 1
  4 0 2 1 2 0 2
  2 2 2 2 0 4 0 0 2 1
  4 1 2 0 6 0 2 2 1 0 2 0 2
  3 2 0 6 0 2 4 4 0 2 0 2 2 0 0 2 1
  4 0 6 0 2 4 5 0 6 0 4 2 0 0 4 1 0 0 2 0 2
  2 4 0 2 6 5 0 6 0 8 4 0 0 6 2 0 2 2 0 2 0 2 0 0 2 1
Row  n = 8 counts the following partitions:
  8          332   53      62       71        521     4211   611      5111
  44               22211   422      2111111   32111          311111   41111
  2222                     431
  11111111                 3221
                           3311
                           221111
(End)

Links

Crossrefs

Cf. A000005, A000041, A002620, A033638, A070824, A188814, A326844, A326849.

Programs

  • Maple
    q := 10: with(combinat): a := proc (i, j) options operator, arrow: partition(i)[j] end proc: P[q] := 0: for j to numbpart(q) do P[q] := sort(P[q]+t^(nops(a(q, j))*max(a(q, j))-q)) end do: P[q] := P[q];
# second Maple program:
b:= proc(n, i, l) option remember; expand(`if`(n=0 or i=1,
      x^(`if`(l=0, 0, n*(l-i))), b(n, i-1, l)+`if`(i>n, 0,
      x^(`if`(l=0, 0, l-i))*b(n-i, i, `if`(l=0, i, l)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 12 2016
  • Mathematica
    b[n_, i_, l_] := b[n, i, l] = Expand[If[n == 0 || i == 1, x^(If[l == 0, 0, n*(l - i)]), b[n, i - 1, l] + If[i > n, 0, x^(If[l == 0, 0, l - i])*b[n - i, i, If[l == 0, i, l]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Max[#]*Length[#]-n==k&]],{n,1,11},{k,0,Floor[(n-1)/2]*Ceiling[(n-1)/2]}] (* Gus Wiseman, Sep 24 2019 *)

Formula

The weight of the complement of a partition p is (number of parts of p)*(largest part of p) - weight of p.
For a given q, the Maple program yields the generating polynomial of row q.

A182094 Total area of the bounding boxes of all integer partitions of n.

Original entry on oeis.org

0, 1, 4, 10, 24, 47, 93, 162, 283, 462, 747, 1154, 1779, 2642, 3908, 5643, 8098, 11398, 15975, 22030, 30253, 41027, 55379, 73983, 98455, 129838, 170578, 222447, 289009, 373064, 479970, 613962, 782893, 993349, 1256546, 1582466, 1987365, 2485840, 3101146
Offset: 0

Views

Author

Alois P. Heinz, Apr 11 2012

Keywords

Examples

			a(4) = 24 = 4+6+4+6+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the bounding boxes have areas 4*1, 3*2, 2*2, 2*3, 1*4.
a(5) = 47 = 5+8+6+9+6+8+5 because the partitions of 5 are [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].

Links

Crossrefs

Cf. A000041, A066186, A115995, A188814.

Programs

  • Maple
    b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    elif i<1 then [0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, n}, If[i < 1, {0, 0}, f = b[n, i - 1]; g = If[i > n, {0, 0}, b[n - i, i]]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := Sum[Sum[i, {i, b[n - j, Min[j, n - j]]}]*j, {j, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)

Formula

a(n) = A188814(n) + n*A000041(n) = A188814(n) + A066186(n).
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