|
| |
|
|
A182395
|
|
Column sums of an infinite Kostka matrix.
|
|
0
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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.
|
|
|
MATHEMATICA
|
(* 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_}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
