A059797 -id:A059797 - cp's OEIS frontend

A067924 Triangle read by rows in which the n-th row gives degrees of irreducible representations of symmetric group S_n (cf. A060240) but now rows are sorted as indicated in A059797 with p(n) terms on each row, where p(n) = A000041(n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 5, 5, 1, 5, 10, 10, 5, 1, 9, 16, 9, 5, 5, 1, 6, 15, 20, 15, 6, 1, 14, 35, 35, 14, 14, 21, 21, 14, 1, 7, 21, 35, 35, 21, 7, 1, 20, 64, 90, 64, 20, 28, 70, 56, 56, 70, 28, 14, 42, 14

Offset: 1

Alford Arnold, Mar 04 2002

These are Betti numbers [Dochtermann]. - N. J. A. Sloane, Dec 02 2015
Number of terms in row n is A000041(n).
Row sums generate sequence A000085: 1, 2, 4, 10, 26, 76, ...
Sum of squares generates A000142; e.g., -1*1 + 4*4 + 6*6 + 4*4 + 1*1 + 5*5 +5*5 = 5! = 120.

			A059797 begins 2, 5, 5, 9, 16, 9, so row six of this sequence begins 1, 5, 10, 10, 5, 1, 9, 16, 9, ...
Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1,  2;
  1, 4,  6,  4,  1,  5, 5;
  1, 5, 10, 10,  5,  1, 9, 16,  9,  5,  5;
  1, 6, 15, 20, 15,  6, 1, 14, 35, 35, 14, 14, 21, 21, 14;
  1, 7, 21, 35, 35, 21, 7,  1, 20, 64, 90, 64, 20, 28, 70, 56, 56, 70, 28, 14, 42, 14;

A. Dochtermann, Face rings of cycles, associahedra, and standard Young tableaux, arXiv preprint arXiv:1503.06243 [math.CO], 2015-2016.

Cf. A000041, A000085, A000142, A059797, A060240.

A167201 Third in a series of triangular subarrays of A117506. Previous arrays are Tables A007318 and A059797.

Original entry on oeis.org

5, 14, 21, 28, 70, 56, 48, 162, 216, 120

Offset: 1

Alford Arnold, Nov 02 2009

This subarray is generated from values related to the source partition 3+3. (cf A161924).

			The domain values begin:
12
20..25
36..41..51
68..73..83..103
so based on function A117506, a(n) begins:
5
14..21
28..70..56
48..162..216..120
Note that A117506(22) maps to Partition 3+3
which corresponds to the 12th natural number appearing in A161924.

A007318 A099627 A117506 A161924

A167202 Fourth in a series of triangular subarrays of A117506. Previous arrays are Tables A007318, A059797 and A167201.

Original entry on oeis.org

5, 21, 14, 56, 70, 28, 120, 216, 162, 48

Offset: 1

Alford Arnold, Nov 04 2009

This subarray is generated from values related to the source partition
2+2+2. (cf A161924). Note that A117506(25) maps to Partition 2+2+2
which corresponds to the 14th natural number appearing in A161924.
Note also that Table A167201 is the transpose of Table a(n) and that
partition 3+3 is the conjugate of partition 2+2+2.

			The A161924 domain values begin:
14
22..29
38..45..59
70..77..91..119
so based on function A117506, a(n) begins:
5
21..14
56..70..28
70..77..91..119

A099627 A007318 A124922 A059797 A161924 A167201 A117506

A124922 Second in a series of triangular arrays providing index numbers for subsequences of A060351.

Original entry on oeis.org

6, 10, 13, 18, 21, 27, 34, 37, 43, 55, 66, 69, 75, 87, 111, 130, 133, 139, 151, 175, 223

Offset: 1

Alford Arnold, Nov 21 2006

The first triangular array is A099627 which provides index numbers in A060351 for Pascal's Triangle (A007318). This second array provides the index numbers in A060351 for array A059797.

Note that this table and A099627 are sub-arrays of table A161924 which has A000041 entries per row. - Alford Arnold, Oct 19 2009

			A060351(34,37,43,55) = (14,35,35,14) = Row Four of Array A059797.

Cf. A007318, A059797, A060351, A099627, A124921.

Cf. A000041, A161924.

I would like a clearer definition of this and other recent triangles from this author. - N. J. A. Sloane, Nov 22 2006

More terms from Alford Arnold, Oct 19 2009

A122932 a(n) = A000085(n) - A000079(n-1).

Original entry on oeis.org

0, 0, 0, 2, 10, 44, 168, 636, 2364, 8984, 34672, 138104, 564408, 2382288, 10333152, 46173968, 211733776, 997182752, 4809439296, 23758139808, 119951644320, 618882541760, 3257839688320, 17492182188992, 95680426983360

Offset: 1

Alford Arnold, Sep 22 2006

Also sums values in A117506 but not in A007318.

			Row five of A117506 is 1 5 9 5 10 16 5 10 9 5 1.
Row five of A007318 is 1 5 10 10 5 1.
So included values are 9 5 16 5 9;
therefore a(5) = 44 = 76 - 32.

Cf. A000079, A000085, A059797, A067924, A117506.

Table[Sum[(2k)!/k!/2^k Binomial[n, 2k], {k, 0, n/2}] - 2^(n - 1) // FullSimplify, {n, 26}] (* Robert G. Wilson v, Sep 27 2006 *)  (* or *)
Table[HypergeometricU[ -(n/2), 1/2, -(1/2)]/(-(1/2))^(-(-n/2)) - 2^(n - 1), {n, 26}] (* Robert G. Wilson v, Sep 27 2006 *)  (* or *)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[NumberOfTableaux[M[Star[n+1]]] - 2^(n - 1), {n, 26}] (* Robert G. Wilson v, Sep 27 2006 *)

a(n) = A000085(n) - A000079(n) for n>0.

More terms from Robert G. Wilson v, Sep 27 2006

A126440 Triangular array read by rows: related to A053445 and A060351 with row sums A000142 (which counts permutations of n objects).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 3, 13, 8, 0, 4, 42, 58, 16, 0, 5, 118, 344, 221, 32

Offset: 1

Alford Arnold, Jan 13 2007

A060351 relates permutations to numeric partitions. Diagonal 1 2 4 8 16 32 ... A000079 sums triangle A007318 and is related to hook partitions. Diagonal 2 13 58 221 ... sums 2 10 34 98 ... related to partition 2+2 3 20 85 ... related to 3+3 4 33 ... related to 4+4 5 ... related to 5+5 etc.

			The array begins
1
0 2
0 2 4
0 3 13 8
0 4 42 58 16
0 5 118 344 221 32

Cf. A000041 A000142 A053445 A059797 A060351 A091307.

A136100 Square each term in the sequence counting standard Young tableaux; cf. A117506.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 4, 9, 1, 1, 16, 25, 36, 25, 16, 1, 1, 25, 81, 25, 100, 256, 25, 100, 81, 25, 1, 1, 36, 196, 196, 225, 1225, 441, 441, 400, 1225, 196, 225, 196, 36, 1, 1, 49, 400, 784, 196, 441, 4096, 4900, 3136, 1764, 1225, 8100, 3136, 4900, 196

Offset: 0

Alford Arnold, Dec 27 2007

When viewed as an irregular table there are A000041 entries per row which sum to 1, 1, 2, 6, 24, 120, 720 ...

			Row five of A117506 is 1 4 5 6 5 4 1 so row five of the present triangle is 1 16 25 36 25 16 1.

Cf. A000041, A000085, A000142, A007318, A008459, A059797, A117506.

a(n) = A117506(n)*A117506(n).

More terms from Alois P. Heinz, Nov 05 2015

A167203 Number of Young tableaux with n cells and k inversions.

Original entry on oeis.org

1, 2, 4, 8, 2, 16, 10, 32, 34, 5, 5, 64, 98, 35, 35, 128, 258, 154, 154, 14, 42, 14

Offset: 1

Alford Arnold, Nov 07 2009

The sequence of rowsums is A000085, and the sequence of number of terms in each row is A002865.

			The irregular triangle begins
..1
..2
..4
..8...2
.16..10
.32..34...5...5
.64..98..35..35
128.258.154.154..14..42..14
etc.

FindStat - Combinatorial Statistic Finder, The number of inversions of a tableau

Cf. A000041, A002865, A000079 (rowsum of A007318), A036799 (rowsum of A059797), A117506.

cp's OEIS Frontend

A067924 Triangle read by rows in which the n-th row gives degrees of irreducible representations of symmetric group S_n (cf. A060240) but now rows are sorted as indicated in A059797 with p(n) terms on each row, where p(n) = A000041(n).

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A167201 Third in a series of triangular subarrays of A117506. Previous arrays are Tables A007318 and A059797.

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A167202 Fourth in a series of triangular subarrays of A117506. Previous arrays are Tables A007318, A059797 and A167201.

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A124922 Second in a series of triangular arrays providing index numbers for subsequences of A060351.

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A122932 a(n) = A000085(n) - A000079(n-1).

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A126440 Triangular array read by rows: related to A053445 and A060351 with row sums A000142 (which counts permutations of n objects).

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A136100 Square each term in the sequence counting standard Young tableaux; cf. A117506.

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Examples

Crossrefs

Formula

Extensions

A167203 Number of Young tableaux with n cells and k inversions.

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