A001255 -id:A001255 - cp's OEIS frontend

A248476 Number of pairs (not necessarily successors) of partitions of n that are incomparable in dominance (natural, majorization) ordering.

0, 0, 0, 0, 0, 4, 8, 30, 70, 170, 340, 770, 1424, 2810, 5166, 9542, 16614, 29596, 49952, 85610, 141604, 234622, 379218, 616008, 976134, 1549134, 2418768, 3771252, 5795300, 8903306, 13497384, 20438432, 30630108, 45789134, 67857566, 100346480, 147170162, 215341690

Offset: 1

Wouter Meeussen, Oct 07 2014

nonn

a(n) is always even since each incomparable pair (p1,p2) has a distinct companion (p2,p1).

Alois P. Heinz, Table of n, a(n) for n = 1..200 (first 55 terms from Wouter Meeussen)
Wikipedia, Dominance Order

Cf. A000041, A001255, A182988, A248475, A265508.

Table[Count[ Flatten[Outer[dominant , Partitions[n], Partitions[n], 1]], 0], {n, 24}] (* see A248475 for definition of 'dominant' *)

a(n) = p(n)^2 - A182988(n), where p(n) denotes the number of partitions of n, A000041(n).

A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024

Table[Binomial[PartitionsP[n],2],{n,0,6}]

a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A366527 Number of integer partitions of 2n containing at least one even part.

Original entry on oeis.org

0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841

Offset: 0

Gus Wiseman, Oct 16 2023

nonn

Also partitions of 2n with even product.

			The a(1) = 1 through a(4) = 16 partitions:
  (2)  (4)    (6)      (8)
       (22)   (42)     (44)
       (211)  (222)    (62)
              (321)    (332)
              (411)    (422)
              (2211)   (431)
              (21111)  (521)
                       (611)
                       (2222)
                       (3221)
                       (4211)
                       (22211)
                       (32111)
                       (41111)
                       (221111)
                       (2111111)

This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

a(n) = A000041(2n) - A000009(2n).

A368565 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) < o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 106, 316, 652, 1388, 2618, 5170, 9164, 16790, 29046, 50714, 84732, 143588, 234048, 385210, 617050, 990868, 1558310, 2459300, 3806838, 5900184

Offset: 1

Clark Kimberling, Dec 31 2023

The definition of d depends on the greedy ordering of the partitions p(i) of n; that is, p(1) >= p(2) >= ... >= p(k), where k = A000041(n); see A366156. The ordinal distance o is defined by o(p(i),p(j)) = |i-j|.

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 0 pairs (p,q) for which d(p,q) < o(p,q), so a(4) = 0.

Cf. A000041, A001255, A366156, A368564, A368566.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + a(n) + A368566(n) = A001255(n) for n >= 1.

A368566 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) > o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

0, 2, 6, 18, 34, 48, 62, 108, 166, 242, 334, 512, 706, 984, 1368, 1876, 2492, 3360, 4422, 5848, 7574, 9792, 12596, 16130, 20412, 25850

Offset: 1

Clark Kimberling, Dec 31 2023

The definition of d depends on the greedy ordering of the partitions p(i) of n; that is, p(1) >= p(2) >= ... >= p(k), where k = A000041(n); see A366156. The ordinal distance o is defined by o(p(i),p(j)) = |i-j|.

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 18 pairs (p,q) for which d(p,q) > o(p,q), so a(4) = 18.

Cf. A000041, A001255, A366156, A368564, A368565.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + A368565(n) + a(n) = A001255(n) for n >= 1.

A369704 Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.

Original entry on oeis.org

1, 1, 2, 4, 8, 13, 28, 43, 84, 137, 243, 372, 684, 1010, 1702, 2620, 4256, 6276, 10134, 14740, 23094, 33742, 51139, 73550, 111303, 158140, 233006, 331099, 481324, 674778, 973928, 1353504, 1925734, 2668263, 3748636, 5153887, 7201684, 9820055, 13572468, 18445878

Offset: 0

Alois P. Heinz, Jan 29 2024

nonn

			a(5) = 13: (11111, 11111), (2111, 11111), (2111, 2111), (2111, 221), (221, 11111), (221, 2111), (221, 221), (311, 11111), (311, 311), (32, 32), (41, 11111), (41, 41), (5, 5).

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A000041, A001255, A077285, A297388, A369707, A369910.

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
      add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..42);

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] + Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], { j, 1, n/i}]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

a(n) = A000041(n) + A369707(n).

A143228 Triangle read by rows, T(n,k) = p(n) * p(k), where p(n) = the number of partitions of n, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 3, 6, 9, 5, 5, 10, 15, 25, 7, 7, 14, 21, 35, 49, 11, 11, 22, 33, 55, 77, 121, 15, 15, 30, 45, 75, 105, 165, 225, 22, 22, 44, 66, 110, 154, 242, 330, 484, 30, 30, 60, 90, 150, 210, 330, 450, 660, 900, 42, 42, 84, 126, 210, 294, 462, 630, 924, 1260, 1764

Offset: 0

Gary W. Adamson, Jul 31 2008

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  4;
   3,  3,  6,  9;
   5,  5, 10, 15, 25;
   7,  7, 14, 21, 35,  49;
  11, 11, 22, 33, 55,  77, 121;
  15, 15, 30, 45, 75, 105, 165, 225;
  ...
T(7,4) = 75 = p(7) * p(4) = 15 * 5.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000041, A143229 (row sums).

Main diagonal gives: A001255.

A143228:= func< n,k | NumberOfPartitions(n)*NumberOfPartitions(k) >;
[A143228(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2024

Table[PartitionsP[n]*PartitionsP[k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2024 *)

def A143215(n,k): return number_of_partitions(n)*number_of_partitions(k)
flatten([[A143215(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 27 2024

T(n, 0) = A000041(n) (left border).
Sum_{k=0..n} T(n, k) = A143229(n) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A000041(n)*A087787(n). - G. C. Greubel, Aug 27 2024

A200089 Decimal expansion of sum of reciprocals of squares of partition numbers.

Original entry on oeis.org

1, 4, 3, 8, 6, 9, 2, 9, 4, 3, 2, 8, 8, 2, 2, 6, 4, 4, 3, 9, 2, 8, 2, 5, 3, 6, 9, 6, 2, 0, 3, 7, 4, 9, 1, 4, 2, 8, 5, 1, 7, 2, 3, 1, 9, 3, 7, 9, 6, 1, 0, 8, 4, 4, 0, 4, 3, 6, 3, 4, 1, 6, 9, 7, 3, 2, 3, 4, 4, 5, 6, 2, 6, 6, 2, 8, 4, 1, 1, 4, 1, 2, 4, 8, 2, 0, 5, 7, 1, 7, 3, 8, 1, 6, 7, 0, 8, 8, 3, 1, 3, 8, 5, 2, 7

Offset: 1

Arkadiusz Wesolowski, Nov 17 2011

Vaclav Kotesovec, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Partition Function P

Cf. A001255.

RealDigits[N[Sum[1/PartitionsP[i]^2, {i, 2632}], 105]][[1]] (* Arkadiusz Wesolowski, Jan 29 2012 *)

Sum_{n=1..inf} 1/(A000041(n))^2 = 1.438692943288226443928253696....

A210966 Sum of all region numbers of all parts of the n-th region of the shell model of partitions.

Original entry on oeis.org

1, 4, 9, 4, 25, 6, 49, 8, 18, 10, 121, 12, 26, 14, 225, 16, 34, 18, 76, 20, 21, 484, 23, 48, 25, 104, 27, 56, 29, 900, 31, 64, 33, 136, 35, 36, 259, 38, 78, 40, 41, 1764, 43, 88, 45, 184, 47, 96, 49, 400, 51, 52, 159, 54, 55, 3136, 57, 116, 59, 240

Offset: 1

Omar E. Pol, Jul 01 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			The first seven regions of the shell model of partitions (or the seven regions of 5) are [1], [2, 1], [3, 1, 1], [2], [4, 2, 1, 1, 1], [3], [5, 2, 1, 1, 1, 1, 1] therefore the "region numbers" are [1], [2, 2], [3, 3, 3], [4], [5, 5, 5, 5, 5], [6], [7, 7, 7, 7, 7, 7, 7]. So a(1)..a(7) give: 1, 4, 9, 4, 25, 6, 49.
Also written as an irregular triangle the sequence begins:
1;
4;
9;
4,25;
6,49;
8,18,10,121;
12,26,14,225;
16,34,18,76,20,21,484;
23,48,25,104,27,56,29,900;
31,64,33,136,35,36,259,38,78,40,41,1764;
43,88,45,184,47,96,49,400,51,52,159,54,55,3136;

Omar E. Pol, Illustration of the seven regions of 5

Row n has length A187219(n). Row sums give A210969. Right border gives A001255, n >= 1.

Cf. A135010, A138121, A194446, A182703, A206437, A210971, A210972.

a(n) = n*A194446(n).

A274691 Number of odd entries in the character table of the symmetric group S_n.

Original entry on oeis.org

1, 1, 4, 7, 19, 33, 77, 135, 218, 392, 798, 1312, 2381, 4107, 6639, 11722, 15869, 26333, 45115, 69168, 106213, 170710, 244042, 384991, 592859, 895944, 1326012, 2055454, 2884762, 4466493, 6553384, 9798596, 13336991, 20192347, 28680574, 41695293, 59766105, 86344867

Offset: 0

Richard Stanley, Jul 02 2016

nonn

			For n = 2, all four character values are 1 or -1, so a(2) = 4.

Hugo Pfoertner, Table of n, a(n) for n = 0..76
Alexander R. Miller, On parity and characters of symmetric groups, preprint.
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231-240. Gives terms for n <= 76.

Cf. A001255, A006907, A335686.

with(combinat):
a:= n-> add(`if`(i[]::odd, 1, 0), i=entries(character(n))):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 10 2016

More terms from Alois P. Heinz, Jul 10 2016

Further terms from Miller (2019) added by N. J. A. Sloane, Jul 07 2020

cp's OEIS Frontend

A248476 Number of pairs (not necessarily successors) of partitions of n that are incomparable in dominance (natural, majorization) ordering.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A355389 Number of unordered pairs of distinct integer partitions of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366527 Number of integer partitions of 2n containing at least one even part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A368565 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) < o(p,q), where d and o are distance functions; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A368566 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) > o(p,q), where d and o are distance functions; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A369704 Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A143228 Triangle read by rows, T(n,k) = p(n) * p(k), where p(n) = the number of partitions of n, for 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A200089 Decimal expansion of sum of reciprocals of squares of partition numbers.

Views

Author

Keywords

Links