A259478 -id:A259478 - cp's OEIS frontend

A297388 Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment).

1, 2, 6, 13, 30, 58, 120, 219, 413, 730, 1296, 2201, 3766, 6206, 10241, 16500, 26502, 41748, 65600, 101417, 156264, 237741, 360146, 539838, 806030, 1192365, 1756766, 2568418, 3739724, 5408247, 7791474, 11156601, 15916288, 22585112, 31933166, 44932450, 63010688

Offset: 0

Richard Stanley, Dec 29 2017

For fixed q, the number of p is given by a determinant due to MacMahon (the case mu=empty set and n=1 of Exercise 3.149 of the reference below).

			For n = 2 the six pairs are (empty set,2), (1,2), (2,2), (empty set,11), (1,11), (11,11).

R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge Univ. Press, 2012.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A259478, A305023.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, 1+
      `if`(t=0, 0, n), b(n, i-1, min(i-1, t))+ add(
       b(n-i, min(i, n-i), min(j, n-i)), j=0..t))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 29 2017

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, 1 + If[t == 0, 0, n], b[n, i - 1, Min[i - 1, t]] + Sum[b[n - i, Min[i, n - i], Min[j, n - i]], {j, 0, t}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) + Sum_{k=1..n} A259478(n,k). - Alois P. Heinz, Jan 10 2018

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.

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}]

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}];

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}]

A303851 Number of pairs (lambda,mu) of partitions lambda of n and mu of floor(n/2) with mu <= lambda (by diagram containment).

Original entry on oeis.org

1, 1, 2, 3, 8, 12, 23, 35, 70, 105, 178, 262, 479, 690, 1119, 1590, 2687, 3756, 5960, 8221, 13203, 17994, 27728, 37363, 58293, 77767, 117084, 154747, 234579, 307400, 455637, 592377, 878603, 1134126, 1652382, 2118344, 3089769, 3936532, 5654741, 7161970

Offset: 0

Alois P. Heinz, May 01 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Bisections give: A303861 (even part), A303862 (odd part).

Cf. A259478.

a(n) = A259478(n,floor(n/2)).

A303852 Number of pairs (lambda,mu) of partitions lambda of n and mu of ceiling(n/2) with mu <= lambda (by diagram containment).

Original entry on oeis.org

1, 1, 2, 4, 8, 13, 23, 42, 70, 114, 178, 313, 479, 759, 1119, 1858, 2687, 4207, 5960, 9468, 13203, 20198, 27728, 42955, 58293, 87333, 117084, 176706, 234579, 346450, 455637, 673619, 878603, 1276936, 1652382, 2404288, 3089769, 4429895, 5654741, 8105634

Offset: 0

Alois P. Heinz, May 01 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Bisections give: A303861 (even part), A303863 (odd part).

Cf. A259478.

a(n) = A259478(n,ceiling(n/2)).

A303861 Number of pairs (lambda,mu) of partitions lambda of 2n and mu of n with mu <= lambda (by diagram containment).

Original entry on oeis.org

1, 2, 8, 23, 70, 178, 479, 1119, 2687, 5960, 13203, 27728, 58293, 117084, 234579, 455637, 878603, 1652382, 3089769, 5654741, 10284636, 18389288, 32649594, 57145095, 99386432, 170658713, 291240275, 491704106, 825049784, 1371281674, 2266239384, 3713251732

Offset: 0

Alois P. Heinz, May 01 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Bisection (even part) of A303851 and of A303852.

Cf. A259478.

a(n) = A259478(2n,n) = A303851(2n) = A303852(2n).

A303862 Number of pairs (lambda,mu) of partitions lambda of 2n+1 and mu of n with mu <= lambda (by diagram containment).

Original entry on oeis.org

1, 3, 12, 35, 105, 262, 690, 1590, 3756, 8221, 17994, 37363, 77767, 154747, 307400, 592377, 1134126, 2118344, 3936532, 7161970, 12955028, 23043927, 40717240, 70937194, 122845251, 210072976, 357119309, 600696846, 1004415874, 1663790776, 2740918749, 4477271509

Offset: 0

Alois P. Heinz, May 01 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Bisection (odd part) of A303851.

Cf. A259478.

a(n) = A259478(2n+1,n) = A303851(2n+1).

A303863 Number of pairs (lambda,mu) of partitions lambda of 2n+1 and mu of n+1 with mu <= lambda (by diagram containment).

Original entry on oeis.org

1, 4, 13, 42, 114, 313, 759, 1858, 4207, 9468, 20198, 42955, 87333, 176706, 346450, 673619, 1276936, 2404288, 4429895, 8105634, 14576610, 26017631, 45767924, 79970030, 137934105, 236375455, 400667547, 674818625, 1125626097, 1866565406, 3068368925, 5015847315

Offset: 0

Alois P. Heinz, May 01 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Bisection (odd part) of A303852.

Cf. A259478.

a(n) = A259478(2n+1,n+1) = A303852(2n+1).

cp's OEIS Frontend

A297388 Number of pairs (p,q) of partitions such that q is a partition of n and p <= q (diagram containment).

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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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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).

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A259479 Skew diagrams, both connected or not.

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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).

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A303851 Number of pairs (lambda,mu) of partitions lambda of n and mu of floor(n/2) with mu <= lambda (by diagram containment).

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A303852 Number of pairs (lambda,mu) of partitions lambda of n and mu of ceiling(n/2) with mu <= lambda (by diagram containment).

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A303861 Number of pairs (lambda,mu) of partitions lambda of 2n and mu of n with mu <= lambda (by diagram containment).

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A303862 Number of pairs (lambda,mu) of partitions lambda of 2n+1 and mu of n with mu <= lambda (by diagram containment).

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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).