A178718 -id:A178718 - 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

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

A182395 Column sums of an infinite Kostka matrix.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 14, 5, 11, 14, 24, 43, 6, 14, 20, 34, 44, 78, 142, 7, 17, 26, 44, 30, 65, 114, 85, 150, 271, 499, 8, 20, 32, 54, 40, 86, 150, 100, 130, 228, 408, 302, 544, 996, 1850, 9, 23, 38, 64, 50, 107, 186, 55, 136, 176, 307, 546, 206, 360, 475, 850, 1543, 633, 1139, 2080, 3846, 7193, 10, 26, 44, 74, 60, 128, 222, 70, 172, 222, 386, 684, 190, 286, 498, 654, 1164, 2100, 336, 772, 1376, 1026, 1838, 3336, 6122, 2474, 4514, 8328, 15518, 29186

Offset: 1

Wouter Meeussen, Apr 28 2012

nonn

The initial terms of the column sums of Kostka matrices of increasing size converge to a(k). As an infinite sequence, a(k) then equals the k-th column sum of an infinite Kostka matrix.
1,
1, 2,
1, 2, 4,
1, 2, 3, 5, 10,
1, 2, 3, 5, 7, 13, 26,
1, 2, 3, 5, 4, 8, 14, 11, 20, 38, 76
1, 2, 3, 5, 4, 8, 14, 10, 13, 23, 42, 32, 60, 116, 232
1, 2, 3, 5, 4, 8, 14,  5, 11, 14, 24, 43, 17, 30, ..
1, 2, 3, 5, 4, 8, 14,  5, 11, 14, 24, 43, 13, 19, ..
...
For column k, and with mu representing the k-th partition of n, it appears that the number of SSYT with contents equal to partition mu becomes constant for n greater than or equal to 2j+2, with j the value for which A000070(j) < k <= A000070(j+1), when the k-th partition of n becomes (k+i, partition_of_k); i >= 0.

			a(7)=14 since the 7th partition of n (n >= 5) is (1^5), (3,1^3), (4,1^3), ... converging to (3+i,1^3); i >= 0. The count of SSYT with content (3+i,1^3) or 3+i ones, and a single 2,3 and 4 is limited to the 14 SSYT
{{432111}} {{42111}{3}} {{43111}{2}} {{43211}{1}} {{4211}{31}}
{{4311}{21}} {{4321}{11}} {{4111}{3}{2}} {{4211}{3}{1}} {{4311}{2}{1}}
{{432}{111}} {{421}{31}{1}} {{431}{21}{1}} {{411}{3}{2}{1}}
extended by i ones in the first row.

(* function 'kostka': see A178718 *)
it=Table[Tr /@ Transpose[ PadLeft[#, PartitionsP[n]] & /@ kostka /@ Partitions[ n ] ], {n, 16}];
First /@ Cases[ Transpose[{PadRight[Part[ it, -2], PartitionsP[16]], Last[ it ]}], {q_,q_}]

A370723 Number of symmetric (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 2, 5, 14, 39, 123, 393, 1352, 4782, 17824, 68481, 274166

Offset: 1

Ludovic Schwob, May 18 2024

			The a(3) = 5 matrices:
  [1 0 0]  [1 0 0]  [0 1 0]  [0 0 1]  [1 1]
  [0 1 0]  [0 0 1]  [1 0 0]  [0 1 0]  [1 0]
  [0 0 1]  [0 1 0]  [0 0 1]  [1 0 0]

Cf. A178718, A135588, A321652, A365961.

a[0] = 1;
a[n_] := a[n] = Length[Select[Subsets[Tuples[Range[n], 2], {n}], Module[{matrix, rows, cols}, matrix = ConstantArray[0, {n, n}]; (matrix[[#[[1]], #[[2]]]] = 1) & /@ #; rows = Total[matrix, {2}]; cols = Total[matrix, {1}]; And[Union[First /@ #] == Range[Max @@ First /@ #], Union[Last /@ #] == Range[Max @@ Last /@ #], Sort[Reverse /@ #] == #, OrderedQ[Reverse[rows]], OrderedQ[Reverse[cols]]]] &]];
Table[a[n], {n, 1, 6}] (* Robert P. P. McKone, May 19 2024, from Gus Wiseman in A135588 *)

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).

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

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A182395 Column sums of an infinite Kostka matrix.

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A370723 Number of symmetric (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica