A182616 -id:A182616 - cp's OEIS frontend

A304710 Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.

0, 0, 0, 1, 2, 6, 12, 25, 46, 85, 146, 250, 410, 666, 1053, 1648, 2527, 3840, 5747, 8525, 12496, 18172, 26165, 37408, 53038, 74714, 104502, 145315, 200808, 276030, 377339, 513342, 694925, 936590, 1256670, 1679310, 2234994, 2963430, 3914701, 5153434, 6760937

Offset: 0

Alois P. Heinz, May 17 2018

nonn

Also the number of partitions of 2n where the number of odd parts in even positions differs from the number of odd parts in odd positions.

			a(3) = 1: the Ferrers-Young diagram of 321 cannot be tiled with dominoes because the numbers of white and black squares (when colored like a chessboard) are different but each domino covers exactly one white and one black square:
   ._____.
   |_|X|_|
   |X|_|
   |_|
.
a(4) = 2: 32111, 521.
a(5) = 6: 3211111, 32221, 4321, 52111, 541, 721.
a(6) = 12: 321111111, 3222111, 33321, 432111, 5211111, 52221, 54111, 543, 6321, 72111, 741, 921.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Mutilated chessboard problem
Wikipedia, Partition (number theory)
Wikipedia, Young tableau, Diagrams
Index entries for sequences related to dominoes

Column k=0 of A304789.

Cf. A000041, A000712, A058696, A144064, A182616, A304662.

b:= proc(n, i, p, c) option remember; `if`(n=0, `if`(c=0, 0, 1),
      `if`(i<1, 0, b(n, i-1, p, c)+b(n-i, min(n-i, i), -p, c+
      `if`(i::odd, p, 0))))
    end:
a:= n-> b(2*n$2, 1, 0):
seq(a(n), n=0..50);
# second Maple program:
a:= n-> (p-> p(2*n)-add(p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..50);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      numtheory[sigma](j)*b(n-j, k), j=1..n)*k/n)
    end:
a:= n-> b(2*n, 1)-b(n, 2):
seq(a(n), n=0..50);

b[n_, i_, p_, c_] := b[n, i, p, c] = If[n == 0, If[c == 0, 0, 1], If[i < 1, 0, b[n, i - 1, p, c] + b[n - i, Min[n - i, i], -p, c + If[OddQ[i], p, 0]]]];
a[n_] := b[2n, 2n, 1, 0];
Table[a[n], {n, 0, 50}]
(* second program: *)
a[n_] := PartitionsP[2n] - Sum[PartitionsP[j]* PartitionsP[n - j], {j, 0, n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

a(n) = A058696(n) - A000712(n) = A000041(2*n) - A000712(n).
a(n) = A144064(2*n,1) - A144064(n,2).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n) * (1 - 2/(3^(1/4)*n^(1/4)) - (sqrt(3)/(2*Pi) + Pi/(48*sqrt(3))) / sqrt(n) + (Pi/(6*3^(3/4)) + 15*3^(1/4)/(8*Pi)) / n^(3/4)). - Vaclav Kotesovec, May 25 2018

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

A365827 Number of strict integer partitions of n whose length is not 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

			The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)

The complement is counted by A140106 shifted left.
Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.
The non-strict version is A058984, complement A004526.
The case not containing n/2 is A365826, non-strict A365825.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A008967, A035363, A078408, A365659.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

a(n) = A000009(n) - A004526(n-1) for n > 0.

A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

Original entry on oeis.org

1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144

Offset: 0

Gus Wiseman, Oct 13 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   19: {8}
   21: {2,4}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   34: {1,7}
   36: {1,1,2,2}

Partitions of this type are counted by A182616, strict A365828.
A066207 lists numbers with all even prime indices, odd A066208.
A086543 lists numbers with at least one odd prime index, counted by A366322.
A300063 ranks partitions of odd numbers.
A366319 ranks partitions of n not containing n/2.
A366321 ranks partitions of 2k that do not contain k.
Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]

A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

Original entry on oeis.org

4, 10, 12, 16, 22, 25, 28, 30, 34, 36, 40, 46, 48, 52, 55, 62, 64, 66, 70, 75, 76, 82, 84, 85, 88, 90, 94, 100, 102, 108, 112, 115, 116, 118, 120, 121, 130, 134, 136, 138, 144, 146, 148, 154, 155, 156, 160, 165, 166, 172, 175, 184, 186, 187, 190, 192, 194, 196

Offset: 1

Gus Wiseman, Oct 16 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   46: {1,9}
   48: {1,1,1,1,2}
   52: {1,1,6}
   55: {3,5}
   62: {1,11}
   64: {1,1,1,1,1,1}

These partitions are counted by A182616, even bisection of A086543.
Not requiring at least one odd part gives A300061.
Allowing partitions of odd numbers gives A366322.
A031368 lists primes of odd index.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&Or@@OddQ/@prix[#]&]

A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

			The a(6) = 1 through a(12) = 7 strict partitions:
  (6)  (7)      (8)      (9)      (10)       (11)       (12)
       (4,2,1)  (5,2,1)  (4,3,2)  (6,3,1)    (5,4,2)    (5,4,3)
                         (5,3,1)  (7,2,1)    (6,3,2)    (7,3,2)
                         (6,2,1)  (4,3,2,1)  (6,4,1)    (7,4,1)
                                             (7,3,1)    (8,3,1)
                                             (8,2,1)    (9,2,1)
                                             (5,3,2,1)  (5,4,2,1)

The second condition alone has bisections A078408 and A365828.
The complement is counted by A365659.
The non-strict version is A365825, complement A238628.
The first condition alone is A365827, complement A140106.
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]

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).

A366532 Heinz numbers of integer partitions with at least one even and odd part.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142

Offset: 1

Gus Wiseman, Oct 16 2023

nonn

These partitions are counted by A006477.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

These partitions are counted by A006477.
Just even: A324929, counted by A047967.
Just odd: A366322, counted by A086543 (even bisection of A182616).
A031368 lists primes of odd index, even A031215.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 lists prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A026804, A244991, A257992.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or@@EvenQ/@prix[#]&&Or@@OddQ/@prix[#]&]

Intersection of A324929 and A366322.

cp's OEIS Frontend

A304710 Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.

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A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

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A365827 Number of strict integer partitions of n whose length is not 2.

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A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

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A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

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A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

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

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A366532 Heinz numbers of integer partitions with at least one even and odd part.

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