A182395 Column sums of an infinite Kostka matrix.

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

Author

Wouter Meeussen, Apr 28 2012

Keywords

• nonn

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.

Examples

			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.
		

Programs

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_}]