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.

Showing 1-4 of 4 results.

A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 42, 82, 114, 195, 258, 466, 587, 954, 1317, 2021, 2637, 4124, 5298, 7995, 10565, 15075, 19665, 28798, 36773, 51509, 67501, 93060, 119299, 165589, 209967, 285535, 366488, 487536, 622509, 833998, 1048119, 1380410, 1754520, 2291406, 2876454
Offset: 0

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(5) = 15 pairs of integer partitions:
      (5)|(5)
     (41)|(5)
     (32)|(5)
    (311)|(5)
    (221)|(5)
    (221)|(32)
   (2111)|(5)
   (2111)|(32)
  (11111)|(5)
  (11111)|(41)
  (11111)|(32)
  (11111)|(311)
  (11111)|(221)
  (11111)|(2111)
  (11111)|(11111)

Links

Crossrefs

Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n, i)*b(n-i, i), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018
  • Mathematica
    Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).
a(n) = A000041(2n) - A362051(n) for n>=1. - Alois P. Heinz, Apr 27 2023

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408
Offset: 1

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
            (111)       (11111)  (222)   (3211)     (3311)    (333)
                                 (3111)  (1111111)  (221111)  (4221)
                                                              (222111)
                                                              (111111111)

Crossrefs

For median instead of mean we have A362558.
The complement is counted by A362560.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

Original entry on oeis.org

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098
Offset: 1

Views

Author

Gus Wiseman, Apr 28 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

Crossrefs

For median instead of mean we have A322439 aerated, complement A362558.
The complement is counted by A362559.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

Original entry on oeis.org

1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

Examples

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)     (7)
            (21)   (31)  (32)     (42)    (43)
            (111)        (41)     (51)    (52)
                         (221)    (222)   (61)
                         (311)    (411)   (322)
                         (2111)   (2211)  (331)
                         (11111)          (421)
                                          (511)
                                          (2221)
                                          (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).

Crossrefs

The odd bisection is A058695.
The version for compositions is A213173.
The complement is counted by A322439 aerated.
The even bisection is A362051.
For mean instead of median we have A362559.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]
Showing 1-4 of 4 results.

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