A259479 -id:A259479 - cp's OEIS frontend

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

1, 1, 2, 2, 4, 5, 8, 4, 11, 12, 16, 12, 32, 28, 31, 8, 64, 31, 128, 33, 82, 64, 256, 28, 69, 144, 69, 86, 512, 105, 1024, 16, 208, 320, 209, 82, 2048, 704, 512, 86, 4096, 318, 8192, 216, 262, 1536, 16384, 64, 465, 262, 1232, 528, 32768, 209, 588, 245, 2912, 3328

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 11 tableaux:
1 1
1 1
.
2 1   1 1   1 1   1 2
1 1   1 2   2 2   1 2
.
1 1   1 2   1 2   1 3
2 3   1 3   3 3   2 3
.
1 2   1 3
3 4   2 4

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]Table[PrimePi[p],{k}]]]];
Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]

A259478 Partition containment triangle.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 8, 7, 5, 7, 12, 13, 12, 7, 11, 20, 23, 25, 19, 11, 15, 28, 35, 42, 39, 30, 15, 22, 42, 54, 70, 70, 66, 45, 22, 30, 58, 78, 105, 114, 119, 99, 67, 30, 42, 82, 112, 158, 178, 202, 186, 155, 97, 42, 56, 110, 154, 223, 262, 313, 314, 292, 226, 139, 56, 77, 152, 215, 319, 383, 479, 503, 511, 442, 336, 195, 77

Offset: 1

Wouter Meeussen, Jun 28 2015

T(n,k) counts pairs of partitions (lambda,mu) with Ferrers diagram of mu not extending beyond the diagram of lambda for all partitions lambda of size n and mu of size k <= n.
First column and main diagonal both equal A000041 (partition numbers).
This sequence counts (2,1)/(1) as different from (3,2,1)/(3,1) while their set-theoretic difference lambda - mu (their skew diagram) is the same.

			T(3,2) = 4, the pairs of partitions are ((3)/(2)), ((2,1)/(2)), ((2,1)/(1,1)), ((1,1,1)/(1,1))
and the diagrams are:
  x x 0 , x x , x 0 , x
          0     x     x
                      0
Triangle begins:
  n=1;  1
  n=2;  2  2
  n=3;  3  4  3
  n=4;  5  8  7  5
  n=5;  7 12 13 12  7
  n=6; 11 20 23 25 19 11

I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Columns k=1-10 give: A000041, 2*A000065, A303853, A303854, A303855, A303856, A303857, A303858, A303859, A303860.

Cf. A000070, A259479, A259480, A259481, A161492, A227309, A006958, A297388, A303851, A303852, A303861, A303862, A303863.

b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
      `if`(t=0, 1, add(x^j, j=0..n)), b(n, i-1, min(i-1, t))+
       add(b(n-i, min(i, n-i), min(j, n-i))*x^j, j=0..t)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$3)):
seq(T(n), n=1..15);  # Alois P. Heinz, Jul 05 2015, revised Jan 10 2018

majorsweak[left_List,right_List]:=Block[{le1=Length[left],le2=Length[right]},If[le2>le1||Min[Sign[left-PadRight[right,le1]]]<0,False,True]];
Table[Sum[ If[! majorsweak[\[Lambda], \[Mu]], 0, 1] , {\[Lambda], IntegerPartitions[n] }, {\[Mu], IntegerPartitions[m]}], {n, 7}, {m, n}]
(* Second program: *)
b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[m > n, 0, If[n == 0, 1, If[i < 1, 0, If[t && j > 0, b[n, m, i, j - 1, t], 0] + If[i > j, b[n, m, i - 1, j, False], 0] + If[i > n || j > m, 0, b[n - i, m - j, i, j, True]]]]]; T[n_, m_] :=  b[n, m, n, m, True]; Table[Table[T[n, m], {m, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Aug 27 2016, after Alois P. Heinz *)

Sum_{k=1..n} T(n,k) = A297388(n) - A000041(n). - Alois P. Heinz, Jan 10 2018

A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 11, 5, 2, 0, 0, 0, 0, 15, 8, 4, 0, 0, 0, 0, 0, 22, 14, 10, 3, 0, 0, 0, 0, 0, 30, 21, 18, 7, 1, 0, 0, 0, 0, 0, 42, 32, 32, 17, 6, 0, 0, 0, 0, 0, 0, 56, 45, 50, 31, 15, 2, 0, 0, 0, 0, 0, 0, 77, 65, 80, 58, 36, 11, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jul 01 2015

In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.
The diagonals T(n,n-k) count connected skew diagrams with weight k:
  1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;
Their sums equal A006958.

			T(7,2) = 4, the pairs of partitions are ((4,3)/(2)), ((3,3,1)/(2)), ((3,2,2)/(1,1)) and ((2,2,2,1)/(1,1));
The diagrams are:
  x x 0 0 , x x 0 , x 0 0 , x 0
  0 0 0     0 0 0   x 0     x 0
            0       0 0     0 0
                            0
Triangle begins:
      k=0  1  2  3  4  5  6  7
  n=0;  0
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  0  0  0
  n=4;  5  1  0  0  0
  n=5;  7  2  0  0  0  0
  n=6; 11  5  2  0  0  0  0
  n=7; 15  8  4  0  0  0  0  0

I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.

Wouter Meeussen, Table n,m, T(n,m) for n= 1..27

Cf. A259478, A259479, A259481, A161492, A227309, A006958.

(* see A259479 *) factor[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=Block[{a1,a2,a3},a1=Apply[Join,Table[{i,j},{i,Length[\[Lambda]]},{j,\[Lambda][[i]],\[Lambda][[Min[i+1,Length[\[Lambda]]]]],-1}]];
a2=Map[{First[#],First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&,a1];a3=Map[First,DeleteCases[SplitBy[a2,MatchQ[#,{,False}]&],{{,False}}],{2}];
Flatten[redu[Part[\[Lambda],#], Part[PadRight[\[Mu],Length[\[Lambda]],0],#]/. 0->Sequence[]]&/@Map[Union,a3],1]];
Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]

A300120 Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

Original entry on oeis.org

2, 6, 12, 26, 44, 86, 136, 239, 376, 613, 930, 1485, 2194, 3355, 4948, 7372, 10656, 15660, 22359, 32308

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(3) = 12 skew partitions:
(3)/()   (3)/(1)   (3)/(2)    (3)/(3)
(21)/()  (21)/(11) (21)/(2)   (21)/(21)
(111)/() (111)/(1) (111)/(11) (111)/(111)

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A297388, A299925, A299926, A300118, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 4, 13, 51, 183, 771, 3087, 13601, 59933, 278797, 1311719, 6453606, 32179898, 166075956, 871713213, 4704669005, 25831172649, 145260890323

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(3) = 13 tableaux:
1 1 1   1 1 2   1 2 2   1 2 3
.
1 1   1 1   1 2   1 2   1 3
1     2     1     3     2
.
1   1   1   1
1   1   2   2
1   2   2   3

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A296561, A297388, A299699, A299925, A299926, A300118, A300120, A300121, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 4, 10, 8, 16, 8, 32, 16, 20, 8, 64, 20, 128, 16, 40, 32, 256, 16, 52, 64, 52, 32, 512, 40, 1024, 16, 80, 128, 104, 40, 2048, 256, 160, 32, 4096, 80, 8192, 64, 104, 512, 16384, 32, 272, 104

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, The a(25) = 52 connected tilings of (3,3).

Cf. A000085, A000898, A056239, A006958, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300060, A300118, A300120, A300121, A300122, A300124.

A300124 Number of ways to tile the diagram of an integer partition of n using connected skew partitions.

Original entry on oeis.org

1, 4, 12, 42, 120, 416, 1184, 3888

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, Connected tilings n = 1..4.
Gus Wiseman, Connected tilings n = 5.
Gus Wiseman, Connected tilings n = 6.

Cf. A000085, A000898, A006958, A138178, A259479, A259480, A296561, A297388, A299926, A300120, A300122, A300123.

A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 7, 5, 8, 7, 9, 6, 8, 7, 10, 6, 10, 8, 10, 7, 11, 8, 12, 6, 9, 9, 11, 8, 13, 10, 10, 7, 14, 9, 15, 8, 11, 11, 16, 7, 15, 11, 11, 9, 17, 11, 12, 8, 12, 12, 18, 9, 19, 13, 12, 7, 13, 10, 20, 10, 13, 12, 21, 9, 22, 14, 15, 11

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(15) = 7 denominators are (), (1), (11), (22), (3), (31), (32) with diagrams:
o o o   . o o   . o o   . . o   . . .   . . .   o o o
o o     o o     . o     . .     o o     . o     o o
Missing are the two disconnected skew partitions:
. . o   . . o
o o     . o

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A297388, A299925, A299926, A300056, A300060, A300120, A300122, A300123, A300124.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 6, 3, 2, 0, 0, 0, 0, 7, 4, 4, 0, 0, 0, 0, 0, 8, 5, 6, 3, 0, 0, 0, 0, 0, 9, 6, 8, 6, 1, 0, 0, 0, 0, 0, 10, 7, 10, 9, 6, 0, 0, 0, 0, 0, 0, 11, 8, 12, 12, 11, 2, 0, 0, 0, 0, 0, 0, 12, 9, 14, 15, 16, 9, 2, 0, 0, 0, 0, 0, 0, 13, 10, 16, 18, 21, 16, 7, 0, 0, 0, 0, 0, 0, 0, 14, 11, 18, 21, 26, 23, 18, 4, 0, 0, 0, 0, 0, 0, 0, 15, 12, 20, 24, 31, 30, 29, 12, 3, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jul 01 2015

Border strips are defined as connected skew tabloids free of 2-by-2 cells.

Row sums are the partition numbers (A000041), diagonals sum to 2^n (A000079).

			T(8,2) = 6, the pairs of partitions are ((5,3)/(2)), ((4,3,1)/(2)), ((4,2,2)/(1,1)), ((3,3,1,1)/(2)), ((3,2,2,1)/(1,1)) and ((2,2,2,1,1)/(1,1)); the diagrams are:
  x x 0 0 0 , x x 0 0 , x 0 0 0 , x x 0 , x 0 0 , x 0
  0 0 0       0 0 0     x 0       0 0 0   x 0     x 0
              0         0 0       0       0 0     0 0
                                  0       0       0
                                                  0
Triangle begins:
      k=0  1  2  3  4  5  6  7
  n=0;  0
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  0  0  0
  n=4;  4  1  0  0  0
  n=5;  5  2  0  0  0  0
  n=6;  6  3  2  0  0  0  0
  n=7;  7  4  4  0  0  0  0  0

I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.

Cf. A259478, A259479, A259480, A161492, A227309, A006958.

(* see A259479 *) Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&( Tr[\[Lambda]]-Tr[\[Mu]]==Length[\[Lambda]]+First[\[Lambda]]-1 )&& redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]

cp's OEIS Frontend

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A259478 Partition containment triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A300120 Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

Views

Author

Keywords

Comments

Links

Crossrefs

A300124 Number of ways to tile the diagram of an integer partition of n using connected skew partitions.

Views

Author

Keywords

Comments

Links

Crossrefs

A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).

Views

Author

Keywords

Comments