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A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank. 6

%I

%S 0,1,1,1,1,1,2,1,2,1,3,1,3,2,1,5,2,4,4,2,1,6,3,5,6,4,2,1,10,5,7,9,7,4,

%T 2,1,13,7,9,11,11,7,4,2,1,19,11,12,15,16,12,7,4,2,1,25,16,15,19,22,18,

%U 12,7,4,2,1,35,24,20,26,29,27,19,12,7,4,2,1

%N Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

%C We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Every partition of n has n ranks. This is a generalization of the Dyson's rank of a partition which is the largest part minus the number of parts. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1.

%C It appears that the sum of the k-th ranks of all partitions of n is equal to zero.

%C Also T(n,k) = number of partitions of n with negative k-th rank.

%C It appears that reversed rows converge to A000070, the same as A208482. - Omar E. Pol, Mar 11 2012

%H Alois P. Heinz, <a href="/A208478/b208478.txt">Rows n = 1..44, flattened</a>

%e For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are

%e ----------------------------------------------------------

%e Partitions First Second Third Fourth

%e of 4 rank rank rank rank

%e ----------------------------------------------------------

%e 4 4-1 = 3 0-1 = -1 0-1 = -1 0-1 = -1

%e 3+1 3-2 = 1 1-1 = 0 0-1 = -1 0-0 = 0

%e 2+2 2-2 = 0 2-2 = 0 0-0 = 0 0-0 = 0

%e 2+1+1 2-3 = -1 1-1 = 0 1-0 = 1 0-0 = 0

%e 1+1+1+1 1-4 = -3 1-0 = 1 1-0 = 1 1-0 = 1

%e ----------------------------------------------------------

%e The number of partitions of 4 with positive k-th ranks are 2, 1, 2, 1 so row 4 lists 2, 1, 2, 1.

%e Triangle begins:

%e 0;

%e 1, 1;

%e 1, 1, 1;

%e 2, 1, 2, 1;

%e 3, 1, 3, 2, 1;

%e 5, 2, 4, 4, 2, 1;

%e 6, 3, 5, 6, 4, 2, 1;

%e 10, 5, 7, 9, 7, 4, 2, 1;

%e 13, 7, 9, 11, 11, 7, 4, 2, 1;

%e 19, 11, 12, 15, 16, 12, 7, 4, 2, 1;

%e 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1;

%e 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1;

%Y Column 1 is A064173. Row sums give A208479.

%Y Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A209616.

%K nonn,tabl

%O 1,7

%A _Omar E. Pol_, Mar 07 2012

%E More terms from _Alois P. Heinz_, Mar 11 2012

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Last modified October 21 11:15 EDT 2019. Contains 328294 sequences. (Running on oeis4.)