A304662 -id:A304662 - cp's OEIS frontend

A300061 Heinz numbers of integer partitions of even numbers.

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 30, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 112, 113, 115, 116, 117, 118, 120

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			75 is the Heinz number of (3,3,2), which has even weight, so 75 belongs to the sequence.
Sequence of even-weight partitions begins: () (2) (1,1) (4) (2,2) (3,1) (2,1,1) (6) (1,1,1,1) (8) (4,2) (5,1) (3,3) (2,2,2) (4,1,1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A300063.

Cf. A000041, A000720, A001222, A056239, A063834, A100118, A112798, A122111, A215366, A296150, A299202, A299757, A300056, A300060, A304662.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
      (i[1])*i[2], i=ifactors(k)[2])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

Select[Range[200],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]

A300063 Heinz numbers of integer partitions of odd numbers.

Original entry on oeis.org

2, 5, 6, 8, 11, 14, 15, 17, 18, 20, 23, 24, 26, 31, 32, 33, 35, 38, 41, 42, 44, 45, 47, 50, 51, 54, 56, 58, 59, 60, 65, 67, 68, 69, 72, 73, 74, 77, 78, 80, 83, 86, 92, 93, 95, 96, 97, 98, 99, 103, 104, 105, 106, 109, 110, 114, 119, 122, 123, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			15 is the Heinz number of (3,2), which has odd weight, so 15 belongs to the sequence.
Sequence of odd-weight partitions begins: (1) (3) (2,1) (1,1,1) (5) (4,1) (3,2) (7) (2,2,1) (3,1,1) (9) (2,1,1,1) (6,1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A300061.

Cf. A000041, A000720, A001222, A056239, A063834, A112798, A215366, A296150, A299757, A300056, A300060, A304662.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
      (i[1])*i[2], i=ifactors(k)[2])::even do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

Select[Range[200],OddQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]

A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Wikipedia, Domino tiling
Gus Wiseman, The a(91) = 5 domino tilings of (6,4).

Cf. A000085, A000720, A000712, A000898, A001222, A004003, A056239, A099390, A138178, A153452, A238690, A296150, A296188, A299925, A299926, A300056, A300061, A304662.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
a:= n-> g(sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..120);  # Alois P. Heinz, May 22 2018

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[Map[Function[x, x-1], l[[Range @ f[[1]]]]], ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]] > 0, k-- ]; If[Length[f] > 0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ @ l[[i]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, l[[1]]], ReplacePart[l, 1 -> Nothing]]], 0];
a[n_] := g[Reverse @ Sort[ Flatten[ Map[ Function[i, Table[PrimePi[i[[1]]], i[[2]]]], FactorInteger[n]]]]];
Array[a, 120] (* Jean-François Alcover, May 28 2018, after Alois P. Heinz *)

A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 107, 108, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131

Offset: 1

Gus Wiseman, Mar 12 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is conjectured to be the Heinz numbers of integer partitions in which the odd parts appear as many times in even as in odd positions.

			Sequence of integer partitions whose Young diagram can be tiled by dominos begins: (), (2), (11), (4), (22), (31), (211), (6), (1111), (8), (42), (51), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Wikipedia, Domino tiling

Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
       (-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
       numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[(-1)^Flatten[Position[primeMS[#],_?OddQ]]]===0&] (* Conjectured *)

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

Original entry on oeis.org

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

A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 1, 6, 2, 2, 2, 10, 3, 4, 1, 2, 6, 14, 4, 6, 4, 4, 0, 2, 2, 12, 22, 5, 8, 7, 6, 2, 4, 4, 0, 0, 4, 1, 2, 25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 46, 44, 7, 12, 17, 14, 8, 8, 8, 0, 4, 12, 5, 6, 0, 8, 2, 0, 8, 4, 0, 4, 0, 0, 0, 2, 2, 0, 0, 4, 1, 2, 0, 0, 2, 0, 1

Offset: 0

Alois P. Heinz, May 18 2018

			T(2,2) = 1: 22.
T(3,0) = 1: 321.
T(3,1) = 6: 111111, 21111, 3111, 411, 51, 6.
T(3,2) = 2: 2211, 42.
T(3,3) = 2: 222, 33.
T(8,36) = 1: 4444.
Triangle T(n,k) begins:
   0,  1;
   0,  2;
   0,  4, 1;
   1,  6, 2,  2;
   2, 10, 3,  4,  1,  2;
   6, 14, 4,  6,  4,  4, 0, 2, 2;
  12, 22, 5,  8,  7,  6, 2, 4, 4, 0, 0, 4, 1, 2;
  25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2;

Alois P. Heinz, Rows n = 0..20, flattened
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

Columns k=0-1 give: A304710, A139582(n) = 2*A000041(n) for n>0.
Row sums give A058696(n) or A000041(2n).
Cf. A000290, A000712, A001105, A048574, A052837, A304662, A304790.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> x^`if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
          `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
seq(T(n), n=0..11);

Sum_{k>0} k * T(n,k) = A304662(n).
T(n,A304790(n)) = 1 for n in { A001105 }.
Sum_{k>=0} T(n,k) = A058696(n) = A000041(2n).
Sum_{k>=1} T(n,k) = A000712(n).
Sum_{k>=2} T(n,k) = A048574(n) = A052837(n).

A296625 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(lambda) with lambda a partition of n.

Original entry on oeis.org

1, 2, 6, 16, 42, 106, 268, 660, 1618, 3922, 9438, 22540, 53528, 126358

Offset: 0

Wouter Meeussen, Dec 17 2017

Diagonal of the matrix formed by products of all pairs of partitions.
Conjecture: a(n) is the number of domino tilings of diagrams of integer partitions of 2n. - Gus Wiseman, Feb 25 2018
The above conjecture is not true, see A304662. - Alois P. Heinz, May 22 2018

			for n=2,
s(2)*s(2) = s(4) + s(3,1) + s(2,2) and
s(1,1) * s(1,1) = s(2,2) + s(2,1,1) + s(1,1,1,1)
for 6 terms in total.

Cf. A296624, A296626, A304662.

Table[Sum[Length[LRRule[\[Lambda], \[Lambda]]], {\[Lambda], Partitions[n]}], {n, 0, 7}];
(* Uses the Mathematica toolbox for Symmetric Functions from A296624. *)

a(n) = A304662(n) for n < 7. - Alois P. Heinz, May 22 2018

a(13)-a(14) from Wouter Meeussen, Nov 22 2018

A304718 Number T(n,k) of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly k horizontally oriented dominoes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 5, 5, 3, 5, 9, 14, 9, 5, 7, 18, 28, 28, 18, 7, 11, 29, 63, 62, 63, 29, 11, 15, 51, 109, 150, 150, 109, 51, 15, 22, 79, 206, 293, 380, 293, 206, 79, 22, 30, 126, 342, 590, 787, 787, 590, 342, 126, 30, 42, 189, 584, 1061, 1675, 1760, 1675, 1061, 584, 189, 42

Offset: 0

Alois P. Heinz, May 17 2018

			:   T(2,0) = 2   :   T(2,1) = 2       :   T(2,2) = 2         :
:   ._.  ._._.   :   .___.  ._.___.   :   .___.  .___.___.   :
:   | |  | | |   :   |___|  | |___|   :   |___|  |___|___|   :
:   |_|  |_|_|   :   | |    |_|       :   |___|              :
:   | |          :   |_|              :                      :
:   |_|          :                    :                      :
:                :                    :                      :
Triangle T(n,k) begins:
   1;
   1,   1;
   2,   2,   2;
   3,   5,   5,    3;
   5,   9,  14,    9,    5;
   7,  18,  28,   28,   18,    7;
  11,  29,  63,   62,   63,   29,   11;
  15,  51, 109,  150,  150,  109,   51,   15;
  22,  79, 206,  293,  380,  293,  206,   79,  22;
  30, 126, 342,  590,  787,  787,  590,  342, 126,  30;
  42, 189, 584, 1061, 1675, 1760, 1675, 1061, 584, 189, 42;
  ...

Alois P. Heinz, Rows n = 0..25, flattened
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

Row sums give A304662.
Main diagonal and column k=0 give A000041.
T(n,floor(n/2)) gives A304719.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od; expand(
        `if`(nops(f)>0 and f[1]>=k, x*h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0))
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
seq(T(n), n=0..12);

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]] ]] - 1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]] > 0, k--]; If[Length[f] > 0 && f[[1]] >= k, x*h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k - 1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i - 1, l] + b[n - i, Min[n - i, i], Append[l, i]]];
T[n_] := CoefficientList[b[2n, 2n, {}], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

T(n,k) = T(n,n-k).

A304677 Total number of tilings of Ferrers-Young diagrams using dominoes and monominoes summed over all partitions of n.

Original entry on oeis.org

1, 1, 4, 9, 27, 60, 170, 377, 996, 2288, 5715, 13002, 32321, 72864, 175137, 400039, 943454, 2133159, 4993737, 11236889, 25995341, 58480330, 133650880, 299347432, 681346296, 1519116099, 3427954877, 7631479391, 17122129103, 37958987956, 84819325972, 187405201004

Offset: 0

Alois P. Heinz, May 16 2018

nonn

Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Partition (number theory)
Wikipedia, Polyomino
Wikipedia, Young tableau, Diagrams

Cf. A304662, A304680.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od; h(subsop(k=1, l), f)+
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> b(n$2, []):
seq(a(n), n=0..23);

A304680 Total number of tilings of Ferrers-Young diagrams using dominoes and at most one monomino summed over all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 6, 23, 16, 76, 42, 239, 106, 688, 268, 1931, 650, 5266, 1580, 13861, 3750, 35810, 8862, 91065, 20598, 226914, 47776, 559271, 109248, 1360152, 248966, 3270429, 562630, 7785974, 1264780, 18378067, 2823958, 43007532, 6282198, 99892837, 13884820

Offset: 0

Alois P. Heinz, May 16 2018

nonn

Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Partition (number theory)
Wikipedia, Polyomino
Wikipedia, Young tableau, Diagrams
Index entries for sequences related to dominoes

Bisection (even part) gives A304662.

Cf. A304677.

h:= proc(l, f, t) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f), t))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(t, h(subsop(k=1, l), f, false), 0)+
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f, t), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f, t), 0)
      fi
    end:
g:= l-> (t-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l),
               is(t, odd))))(add(i, i=l)):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> b(n$2, []):
seq(a(n), n=0..23);

cp's OEIS Frontend

A300061 Heinz numbers of integer partitions of even numbers.

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A300063 Heinz numbers of integer partitions of odd numbers.

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A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

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A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

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

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A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

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A296625 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(lambda) with lambda a partition of n.

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A304718 Number T(n,k) of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly k horizontally oriented dominoes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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