A259480 -id:A259480 - 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}]

A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 4, 6, 7, 6, 7, 7, 9, 9, 5, 8, 9, 9, 10, 12, 11, 10, 9, 10, 13, 10, 13, 11, 14, 12, 6, 15, 15, 14, 12, 13, 17, 18, 13, 14, 19, 15, 16, 16, 19, 16, 11, 15, 16, 21, 19, 17, 14, 18, 17, 24, 21, 18, 19, 19, 23, 22, 7, 22, 24, 20, 22, 27, 23, 21

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Equivalently, a(n) equals the number of values of m such that each value of A238689 T(m,k) <= A238689 T(n,k). (Since the prime factorization of 1 is the empty factorization, we consider each prime_1(i) not to be greater than prime_n(i) for all positive integers n.)
Suppose we say that n "covers" m iff both m and n are factorized as described in the sequence definition and each prime_m(i) <= prime_n(i). At least three sequences (A037019, A108951 and A181821) have the property that a(m) divides a(n) iff n "covers" m.  These sequences are also divisibility sequences (i.e., sequences with the property that a(m) divides a(n) if m divides n), since any positive integer "covers" each of its divisors.
For any positive integers m and k, the following integer sequences (with n >= 0) are arithmetic progressions:
1. The sequence b(n) = a(m*(2^n)).
2. The sequence b(n) = a(m*(prime(n+k))) if prime(k) >= A006530(m).
Also, a(n) = the number of distinct prime signatures that occur among the divisors of any integer m such that A181819(m) = n and/or A238745(m) = n.
Number of skew partitions whose numerator has Heinz number n, where a skew partition is a pair y/v of integer partitions such that the diagram of v fits inside the diagram of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 24 2018

			The prime factorizations of integers 1 through 9, with prime factors sorted from largest to smallest:
1 - the empty factorization (no prime factors)
2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 3*2
7 = 7
8 = 2*2*2
9 = 3*3
To find a(9), we consider 9 = 3*3. There are 6 positive integers (1, 2, 3, 4, 6 and 9) which satisfy the following criteria:
1) The largest prime factor, if one exists, is not greater than 3;
2) The second-largest prime factor, if one exists, is not greater than 3;
3) The total number of prime factors (counting repeated factors) does not exceed 2.
Therefore, a(9) = 6.
From _Gus Wiseman_, Feb 24 2018: (Start)
Heinz numbers of the a(15) = 9 partitions contained within the partition (32) are 1, 2, 3, 4, 5, 6, 9, 10, 15. The a(15) = 9 skew partitions are (32)/(), (32)/(1), (32)/(11), (32)/(2), (32)/(21), (32)/(22), (32)/(3), (32)/(31), (32)/(32).
Corresponding diagrams are:
  o o o   . o o   . o o   . . o   . . o   . . o   . . .   . . .   . . .
  o o     o o     . o     o o     . o     . .     o o     . o     . .    (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rearrangement of A115728, A115729 and A238746. A116473(n) is the number of times n appears in the sequence.

Cf. A000041, A000085, A000720, A056239, A063834, A112798, A122111, A153452, A215366, A238689, A259478, A259480, A296150, A296188, A296561, A297388, A299925, A299926, A299966, A299967.

undptns[y_]:=Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[undptns[Reverse[primeMS[n]]]],{n,100}] (* Gus Wiseman, Feb 24 2018 *)

a(n) = A085082(A108951(n)) = A085082(A181821(n)).
a(n) = a(A122111(n)).
a(prime(n)) = a(2^n) = n+1.
a((prime(n))^m) = a((prime(m))^n) = binomial(n+m, n).
a(A002110(n)) = A000108(n+1).
A000005(n) <= a(n) <= n.

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

A259479 Skew diagrams, both connected or not.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 5, 2, 0, 0, 0, 11, 9, 6, 1, 0, 0, 0, 15, 13, 12, 6, 0, 0, 0, 0, 22, 20, 22, 14, 3, 0, 0, 0, 0, 30, 28, 36, 27, 13, 2, 0, 0, 0, 0, 42, 40, 56, 48, 31, 11, 1, 0, 0, 0, 0, 56, 54, 82, 77, 59, 33, 9, 0, 0, 0, 0, 0, 77, 75, 120, 121, 106, 72, 30, 6, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jun 28 2015

T(n,m) counts pairs of partitions lambda of n and mu of 0<=m<=n respectively, so that the Ferrers diagram of mu does not exceed that of lambda, and that the diagrams of lambda and mu do not contain equal rows or columns.

			T(6,2) = 6, the pairs of partitions are ((4,2)/(2)), ((3,3)/(2)), ((3,2,1)/(2)), ((3,2,1)/(1,1)), ((2,2,2)/(1,1)) and ((2,2,1,1)/(1,1))
and the diagrams are:
  x x 0 0 , x x 0 , x x 0 , x 0 0 , x 0 , x 0
  0 0       0 0 0   0 0     x 0     x 0   x 0
                    0       0       0 0   0
                                          0
Triangle begins:
      k=0  1  2  3  4  5  6
  n=0;  1
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  1  0  0
  n=4;  5  3  0  0  0
  n=5;  7  5  2  0  0  0
  n=6; 11  9  6  1  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, A259480, A259481, A161492, A227309, A006958.

majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
redu1[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=Delete[#,List/@DeleteCases[Table[i Boole[\[Lambda][[i]]==\[Mu][[i]]],{i,Length[\[Mu]]}],0]]&/@{\[Lambda],\[Mu]};
redu[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=TransposePartition/@Apply[redu1,TransposePartition/@redu1[\[Lambda],\[Mu]]];
Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[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.

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A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)*prime_m(2)*...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

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A259478 Partition containment triangle.

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A259479 Skew diagrams, both connected or not.

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

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

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Mathematica

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

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

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

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A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).