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Sum of all parts minus the total number of parts of the shell model of partitions with n regions.
2

%I #27 Oct 16 2024 06:03:28

%S 0,1,3,4,8,10,15,16,20,22,31,33,38,41,51,52,56,58,67,71,74,90,92,97,

%T 100,110,112,119,123,142,143,147,149,158,162,165,181,184,192,197,201,

%U 228,230,235,238,248,250,257,261,280,284,287,299,305,310,341

%N Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

%C For the definition of "region of n" see A206437.

%F a(n) = A182244(n) - A182181(n).

%F a(A000041(n)) = A196087(n).

%e Written has a triangle:

%e 0,

%e 1,

%e 3,

%e 4, 8;

%e 10, 15;

%e 16, 20, 22, 31;

%e 33, 38, 41, 51;

%e 52, 56, 58, 67, 71, 74, 90;

%e 92, 97,100,110,112,119,123,142;

%e 143,147,149,158,162,165,181,184,192,197,201,228;

%e 230,235,238,248,250,257,261,280,284,287,299,305,310,341;

%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];

%t reg = {}; l = {};

%t For[j = 1, j <= 56, j++,

%t mx = Max@lex[j][[j]]; AppendTo[l, mx];

%t For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];

%t t = Take[Reverse[First /@ lex[mx]], j - i];

%t AppendTo[reg, Total@t - Length@t]

%t ];

%t Accumulate@reg (* _Robert Price_, Jul 25 2020 *)

%Y Row j has length A187219(j). Right border gives A196087.

%Y Cf. A000041, A066186, A006128, A135010, A138121, A182181, A182244, A186114, A206437, A207035.

%K nonn

%O 1,3

%A _Omar E. Pol_, Apr 23 2012