A097522 -id:A097522 - cp's OEIS frontend

A104778 Table of values with shape sequence A000041 related to involutions and multinomials. Also column sums of the Kostka matrices associated with the partitions (in Abramowitz & Stegun ordering).

1, 1, 1, 2, 1, 2, 4, 1, 2, 3, 5, 10, 1, 2, 3, 5, 7, 13, 26, 1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76, 1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232, 1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764, 1

Offset: 0

Alford Arnold, Mar 24 2005

Row sums give A178718.

			The 47 multinomials (corresponding to A005651(4)=47) can be distributed as in the following triangular array:
  1
  9 1
  4 6 1
  9 2 3 1
  1 3 2 3 1
divide each term by
  1
  3 1
  2 3 1
  3 2 3 1
  1 3 2 3 1
yielding
  1
  3 1
  2 2 1
  3 1 1 1
  1 1 1 1 1
with column sums 10 5 3 2 1.
Therefore the fourth row of the table is 1 2 3 5 10
The initial rows are:
  1,
  1,
  1, 2,
  1, 2, 4,
  1, 2, 3, 5, 10,
  1, 2, 3, 5, 7, 13, 26,
  1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76,
  1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232,
  1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764,
  ...

Wouter Meeussen, Table of n, a(n) for n = 0..372
Wouter Meeussen, Table of n, a(n), partition parts for n = 0..372

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A104778, A178718.

A001475 and A000085 are subsequences.

(* for function 'kostka' see A178718 *)
aspartitions[n_] := Reverse /@ Sort[Sort /@ Partitions[n]];
asorder[n_] := rankpartition /@ Reverse /@ Sort[Sort /@ Partitions[n]];
Flatten[Table[Tr/@ Transpose[PadLeft[#,PartitionsP[k]] [[asorder[k]] ]&/@ kostka/@ aspartitions[k]],{k,11}]]

Corrected and edited by Wouter Meeussen, Jan 15 2012

A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 25, 1, 81, 20, 1, 25, 54, 15, 1, 100, 15, 36, 15, 1, 256, 60, 10, 27, 10, 1, 25, 128, 30, 5, 27, 10, 1, 100, 10, 64, 30, 5, 18, 10, 1, 81, 40, 5, 32, 10, 5, 9, 5, 1, 25, 27, 10, 0, 32, 10, 0, 9, 5, 1, 1, 5, 9, 10, 5, 16, 10, 5, 9, 5, 1

Offset: 1

Alford Arnold, Mar 19 2005

The last row (and the square roots of the first column) is 1 5 9 5 10 16 5 10 9 5 1, which when sorted appears as 1 1 5 5 5 5 9 9 10 10 16 in A003870 and A060240. The 1602 cases are distributed in A036038: 1 6 15 20 30 60 90 120 180 360 720.

			The triangle is:
1;
25,   1;
81,  20,  1;
25,  54, 15,  1;
100, 15, 36, 15,  1;
256, 60, 10, 27, 10,  1;
25, 128, 30,  5, 27, 10,  1;
100, 10, 64, 30,  5, 18, 10, 1;
81,  40,  5, 32, 10,  5,  9, 5, 1;
25,  27, 10,  0, 32, 10,  0, 9, 5, 1;
1,    5,  9, 10,  5, 16, 10, 5, 9, 5, 1;

D. Stanton and D. White, Constructive Combinatorics, 1986, page 83.

Cf. A000041, A000085, A000142, A003870, A060240, A005651, A036038 and A097522 (a similar triangle distributing the 246 multinomials in A005651(5)).

Cf. A104778.

A104779 a(n) is the sum of entries of n-th Kostka matrix for the partitions of n.

Original entry on oeis.org

1, 1, 3, 7, 21, 57, 182, 565, 1931, 6670, 24537, 92337, 364602, 1477148, 6219031, 26875932, 119930947, 548688443, 2580814003, 12425175838, 61302331782, 309055818656, 1592723862598, 8374123173858, 44917765035082, 245452258746785, 1366116578058731, 7736098938006873

Offset: 0

Alford Arnold, Mar 24 2005

a(n) is the number of symmetric nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and weakly decreasing row and column sums. - Ludovic Schwob, Aug 29 2023

			For n=4, {1,1,1,1,1} + {0,1,1,2,3} + {0,0,1,1,2} + {0,0,0,1,3} + {0,0,0,0,1} = 21.

Ludovic Schwob, Table of n, a(n) for n = 0..39
E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix
E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix, European Journal of Combinatorics, Volume 31, Issue 8, December 2010, Pages 2014-2027.
Wouter Meeussen, Schur Polynomials
Wouter Meeussen, Kostka numbers up to partitions of 20
Wouter Meeussen, Mathematica code for 'kostka' function
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A178718.

Row sums of A104778.

Mathematica
```
(* See Meeussen link. *)
```

Row sums of A104778.

a(7) corrected by Alford Arnold, Dec 31 2010

a(8)-a(21) from Amiram Eldar, May 03 2024

A115351 Sum of interior Multinomial Coefficient components.

Original entry on oeis.org

1, 1, 0, 0, 11, 96, 798, 6197, 54400, 505503, 5241223, 58377002, 712436696, 9315437345, 131487856629, 1978064399766, 31777977184459, 541010185315536, 9758067888585784, 185538235462354828, 3714549428287398782

Offset: 0

Alford Arnold, Jan 20 2006

nonn

For a given value of n, the multinomial coefficients can be decomposed into components arranged in triangular fashion, as illustrated in A097522 and A104707. The values on the three edges sum to A000142(n), A000085(n) and A000041(n) respectively. Since each vertex component has the value one and appears on two of the three edges the formula is adjusted by three.

			a(5) = 96 because the sum for the below triangle is 246 and the three edges sum to 120, 26 and 7; therefore 246 - (120 + 26 + 7 - 3) = 96.
1
16 1
25 12 1
36 15 8 1
25 18 10 8 1
16 10 6 5 4 1
1 4 5 6 5 4 1

Cf. A000041, A000085, A000142, A005651, A097522, A104778.

a(n) = A005651(n) - A000142(n) - A000085(n) - A000041(n) + 3

More terms from R. J. Mathar, Jan 23 2008

cp's OEIS Frontend

A104778 Table of values with shape sequence A000041 related to involutions and multinomials. Also column sums of the Kostka matrices associated with the partitions (in Abramowitz & Stegun ordering).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A104779 a(n) is the sum of entries of n-th Kostka matrix for the partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A115351 Sum of interior Multinomial Coefficient components.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions