This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A208478 #53 Aug 21 2021 13:15:23 %S A208478 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 A208478 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 A208478 12,7,4,2,1,35,24,20,26,29,27,19,12,7,4,2,1 %N A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank. %C A208478 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 A208478 The sum of the k-th ranks of all partitions of n is equal to zero. %C A208478 Also T(n,k) = number of partitions of n with negative k-th rank. %C A208478 It appears that reversed rows converge to A000070, the same as A208482. - _Omar E. Pol_, Mar 11 2012 %C A208478 From _Omar E. Pol_, Dec 12 2019: (Start) %C A208478 1) The k-th part of a partition of n is also the number of parts >= k of its conjugate partition. %C A208478 2) The k-th rank of a partitions is also the number of parts >= k of its conjugate partition minus the number of parts >= k. %C A208478 For example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. The mentioned conjugate partition is [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. the Third part is 2, so the conjugate partition has two parts >= 3, and so on. In this case the conjugate partition is [5, 3, 1]. %C A208478 3) The difference between the k-th part and the (k+1)-st part of the partition of n is also the number of k's in its conjugate partition. For example: consider the partition [5, 3, 1]. The difference between the first and the second part is 5 - 3 = 2, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 3 - 1 = 2, equals the number of 2's in its conjugate partition. The difference between the third and the fourth (virtual) part is 1 - 0 = 1, equals the number of 3's in its conjugate partition [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The difference between the first and the second part is 3 - 2 = 1, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 2 - 2 = 0, equals the number of 2's in its conjugate partition. The difference between the third and the fourth part is 2 - 1 = 1, equals the number of 3's in its conjugate partition, and so on. %C A208478 4) The list of n ranks of a partition of n equals the list of n ranks multiplied by -1 of its conjugate partition. For example the nine ranks of the partition [5, 3, 1] of 9 are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0]. %C A208478 For a list of partitions of the positive integers ordered by its k-th ranks see A330370. (End) %H A208478 Alois P. Heinz, <a href="/A208478/b208478.txt">Rows n = 1..44, flattened</a> %e A208478 For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are %e A208478 ---------------------------------------------------------- %e A208478 Partitions First Second Third Fourth %e A208478 of 4 rank rank rank rank %e A208478 ---------------------------------------------------------- %e A208478 4 4-1 = 3 0-1 = -1 0-1 = -1 0-1 = -1 %e A208478 3+1 3-2 = 1 1-1 = 0 0-1 = -1 0-0 = 0 %e A208478 2+2 2-2 = 0 2-2 = 0 0-0 = 0 0-0 = 0 %e A208478 2+1+1 2-3 = -1 1-1 = 0 1-0 = 1 0-0 = 0 %e A208478 1+1+1+1 1-4 = -3 1-0 = 1 1-0 = 1 1-0 = 1 %e A208478 ---------------------------------------------------------- %e A208478 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 A208478 Triangle begins: %e A208478 0; %e A208478 1, 1; %e A208478 1, 1, 1; %e A208478 2, 1, 2, 1; %e A208478 3, 1, 3, 2, 1; %e A208478 5, 2, 4, 4, 2, 1; %e A208478 6, 3, 5, 6, 4, 2, 1; %e A208478 10, 5, 7, 9, 7, 4, 2, 1; %e A208478 13, 7, 9, 11, 11, 7, 4, 2, 1; %e A208478 19, 11, 12, 15, 16, 12, 7, 4, 2, 1; %e A208478 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1; %e A208478 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1; %e A208478 ... %Y A208478 Column 1 is A064173. %Y A208478 Row sums give A208479. %Y A208478 Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370. %K A208478 nonn,tabl %O A208478 1,7 %A A208478 _Omar E. Pol_, Mar 07 2012 %E A208478 More terms from _Alois P. Heinz_, Mar 11 2012