cp's OEIS Frontend

A182395 Column sums of an infinite Kostka matrix.

%I A182395 #21 Apr 15 2023 15:33:57
%S A182395 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,
%T A182395 114,85,150,271,499,8,20,32,54,40,86,150,100,130,228,408,302,544,996,
%U A182395 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
%N A182395 Column sums of an infinite Kostka matrix.
%C A182395 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.
%C A182395 1,
%C A182395 1, 2,
%C A182395 1, 2, 4,
%C A182395 1, 2, 3, 5, 10,
%C A182395 1, 2, 3, 5, 7, 13, 26,
%C A182395 1, 2, 3, 5, 4, 8, 14, 11, 20, 38, 76
%C A182395 1, 2, 3, 5, 4, 8, 14, 10, 13, 23, 42, 32, 60, 116, 232
%C A182395 1, 2, 3, 5, 4, 8, 14,  5, 11, 14, 24, 43, 17, 30, ..
%C A182395 1, 2, 3, 5, 4, 8, 14,  5, 11, 14, 24, 43, 13, 19, ..
%C A182395 ...
%C A182395 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.
%e A182395 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
%e A182395 {{432111}} {{42111}{3}} {{43111}{2}} {{43211}{1}} {{4211}{31}}
%e A182395 {{4311}{21}} {{4321}{11}} {{4111}{3}{2}} {{4211}{3}{1}} {{4311}{2}{1}}
%e A182395 {{432}{111}} {{421}{31}{1}} {{431}{21}{1}} {{411}{3}{2}{1}}
%e A182395 extended by i ones in the first row.
%t A182395 (* function 'kostka': see A178718 *)
%t A182395 it=Table[Tr /@ Transpose[ PadLeft[#, PartitionsP[n]] & /@ kostka /@ Partitions[ n ] ], {n, 16}];
%t A182395 First /@ Cases[ Transpose[{PadRight[Part[ it, -2], PartitionsP[16]], Last[ it ]}], {q_,q_}]
%K A182395 nonn
%O A182395 1,2
%A A182395 _Wouter Meeussen_, Apr 28 2012