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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 column sums 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 or equal than 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

Table of n, a(n) for n=1..97.

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

Sequence in context: A120255 A245608 A244154 * A244983 A059450 A255972

Adjacent sequences:  A182392 A182393 A182394 * A182396 A182397 A182398

KEYWORD

nonn

AUTHOR

Wouter Meeussen, Apr 28 2012

STATUS

approved

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Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)