cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A119559 Inverse Euler transform of A119268.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 6, 12, 24, 49, 108, 239, 554, 1311, 3200
Offset: 0

Views

Author

Franklin T. Adams-Watters, May 30 2006

Keywords

Comments

Is this sequence always nonnegative? If so, is there a combinatorial interpretation?

Programs

  • Mathematica
    A119268 = Cases[Import["https://oeis.org/A119268/b119268.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{0}, EulerInvTransform[Rest @ A119268]] (* Jean-François Alcover, Feb 23 2020, updated Mar 17 2020 *)

A119269 Table by antidiagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 6, 6, 4, 2, 1, 1, 1, 8, 11, 7, 4, 2, 1, 1, 1, 12, 19, 13, 7, 4, 2, 1, 1, 1, 16, 33, 25, 14, 7, 4, 2, 1, 1, 1, 22, 55, 49, 27, 14, 7, 4, 2, 1, 1, 1, 29, 95, 93, 55, 28, 14, 7, 4, 2, 1, 1, 1, 40, 158, 181, 111, 57, 28, 14, 7, 4, 2, 1, 1
Offset: 1

Views

Author

Franklin T. Adams-Watters, May 11 2006

Keywords

Comments

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.
Transposed table is A119338. - Max Alekseyev, May 14 2006

Examples

			Table starts:
  1, 1,  1,  1,  1
  1, 1,  1,  1,  1
  1, 2,  2,  2,  2
  1, 3,  4,  4,  4
  1, 4,  6,  7,  7
  1, 6, 11, 13, 14

Crossrefs

Columns A005987, A000786, A119266, A119267; diagonal A119268. Cf. A096751, A119270.
Cf. A119339, A119340, A119341, A119342.

Formula

a(n,m) = a(n,n-2) for m >= n-1.

Extensions

More terms from Max Alekseyev, May 14 2006

A119338 Table by antidiagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 4, 1, 1, 1, 2, 4, 6, 6, 1, 1, 1, 2, 4, 7, 11, 8, 1, 1, 1, 2, 4, 7, 13, 19, 12, 1, 1, 1, 2, 4, 7, 14, 25, 33, 16, 1, 1, 1, 2, 4, 7, 14, 27, 49, 55, 22, 1, 1, 1, 2, 4, 7, 14, 28, 55, 93, 95, 29, 1, 1, 1, 2, 4, 7, 14, 28, 57, 111, 181, 158, 40, 1
Offset: 1

Views

Author

Max Alekseyev, May 15 2006

Keywords

Comments

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Examples

			Table starts:
  1, 1, 1, 1, 1,  1, ...
  1, 1, 2, 3, 4,  6, ...
  1, 1, 2, 4, 6, 11, ...
  1, 1, 2, 4, 7, 13, ...
  1, 1, 2, 4, 7, 14, ...
  ...

Crossrefs

Rows: A000012, A046682, A000786, A119266, A119267, A119340, A119341, A119342 stabilize to A119268. Transposed table is A119269. Cf. A119339, A119270, A118364, A118365.

A118364 Limit of the number of exactly m-dimensional partitions of m+n as m tends to infinity.

Original entry on oeis.org

0, 1, 2, 6, 19, 60
Offset: 1

Views

Author

Max Alekseyev, May 16 2006

Keywords

Comments

Partial sums are given by A118365.

Crossrefs

Cf. A119270, A119339, A119268.

Formula

a(n)=A119270(m+n,m)=A119339(m+n,n) for all m>=2n-5

A119340 Number of 5-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 57, 117, 251, 543, 1209, 2724, 6251, 14505, 34055, 80450, 191166, 455473, 1086863, 2592817
Offset: 0

Views

Author

Max Alekseyev, May 16 2006

Keywords

Comments

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Crossrefs

Cf. A000390, A119266, A119267, A119268, A119269, A119338, A119341, A119342.

Extensions

a(18)-a(21) from Max Alekseyev, Mar 30 2025

A119341 Number of 6-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 58, 119, 257, 562, 1268, 2910, 6844, 16371, 39910, 98667, 247200, 625559
Offset: 0

Views

Author

Max Alekseyev, May 16 2006

Keywords

Comments

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Crossrefs

Cf. A000416, A119266, A119267, A119268, A119269, A119338, A119340, A119342.

Extensions

a(16)-a(19) from Max Alekseyev, Mar 31 2025

A119342 Number of 7-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 58, 120, 259, 568, 1287, 2970, 7036, 17009, 42042, 105848
Offset: 0

Views

Author

Max Alekseyev, May 16 2006

Keywords

Comments

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Crossrefs

Cf. A000427, A119266, A119267, A119268, A119269, A119338, A119340, A119341.

Extensions

a(15)-a(17) from Max Alekseyev, Apr 02 2025

A118365 Limiting difference of the number of infinity-dimensional partitions and m-dimensional partitions of m+n as m tends to infinity.

Original entry on oeis.org

0, 1, 3, 9, 28, 88
Offset: 2

Views

Author

Max Alekseyev, May 17 2006

Keywords

Crossrefs

Cf. A119269, A119338, A119268.

Formula

a(n)=A119268(m+n)-A119269(m+n,m)=A119268(m+n)-A119338(m,m+n) for all m>=2n-8. Partial sums of A118364.

A226651 Multidimensional Young numbers: Given a d-dimensional partition of n, this is the number of ways to fill the associated d-dimensional Young diagram with the integers 1 to n such that the entries are increasing in each positive (orthogonal) direction.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 6, 1, 4, 5, 6, 12, 8, 24, 1, 5, 9, 10, 5, 16, 20, 25, 30, 20, 16, 60, 40, 120, 1, 6, 14, 15, 14, 25, 20, 21, 30, 54, 60, 30, 96, 40, 66, 61, 75, 90, 48, 120, 150, 180, 120, 96, 80, 360, 240, 720, 1, 7, 20, 21, 28, 64, 35, 14, 70, 56, 90, 42, 42, 98, 105, 98, 245, 140, 147
Offset: 1

Views

Author

Graham H. Hawkes, Jul 30 2013

Keywords

Comments

Generalization of the number of standard Young tableaux on a given Young diagram to arbitrary dimension.
The multidimensional Young numbers of partitions which are conjugate are equal. Therefore, the multidimensional Young numbers listed above are indexed with respect to an ordering of the "conjugacy classes" of partitions. This ordering is defined in the attached pdf file.
The number of entries between the m-th and (m+1)-th appearance of 1 (including the m-th appearance, but excluding the (m+1)-th) is the number of infinite dimensional partitions of m up to conjugacy, i.e., sequence A119268.
Let f(m) give the number of 2-dimensional partitions of m up to conjugacy (sequence A005987). Then the first f(m) entries following (and including) the m-th appearance of 1 are standard Young tableaux numbers on 2-dimensional partitions of m, and can be found in sequence A117506.

Examples

			The ordering of "conjugacy classes" of partitions begins:
(1), (2), (3), (2+1), (4), (3+1), (2+2), ((2+1)+(1)), (5), (4+1), (3+2), (3+1+1), ((3+1)+(1)), ((2+2)+(1)),
  (((2+1)+(1))+((1))), ...
The 14th partition, ((2+2)+(1)), is associated to the Young diagram with cubes centered at p_1=(0,0,0), p_2=(1,0,0), p_3=(0,1,0), p_4=(1,1,0), and p_5=(0,0,1). The possible ways to fill the cubes centered on these points so that the numbers are increasing in all directions are;
(For each i=1:5, the i-th integer in a sequence below is placed on p_i.)
1-2-3-4-5
1-3-2-4-5
1-2-3-5-4
1-3-2-5-4
1-2-4-5-3
1-4-2-5-3
1-3-4-5-2
1-4-3-5-2
Hence the 14th term is 8.
The 48th partition, ((2+2)+(2+2)), can be represented as a cube divided into octants. The integers 1 and 8 must lie in opposite octants. Of the three octants adjacent to the one which contains 1, one must contain 2 and one must contain 3. This gives 6 possibilities. For each of these possibilities there are 4 numbers (4, 5, 6, and 7) to choose from for the number placed in the remaining cube in the plane that contains 1, 2, and 3. Regardless of this choice, there are 2 ways to fill in the remaining three octants. Thus there are 6*4*2=48 ways to fill the octants all together--that is, the 48th multidimensional Young number is 48.
Example of recursion:
The partition: p|--6=((3+2)+(1)) covers the following partitions of 5:
q_1|--5=(3+2)
q_2|--5=((3+1)+(1))
q_3|--5=((2+2)+(1)) Thus Y(p)=Y(q_1)+Y(q_2)+Y(q_3)=5+12+8=25

Links

Programs

  • MATLAB
    % See MATLAB function in Links.

Formula

Let p be a partition of n. Let Q be the set of partitions of n-1 such that for all q in Q, p covers q. Then the Young number of p is given by Y(p) = Sum_{q in Q} Y(q).
Showing 1-9 of 9 results.

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