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 A330370 #132 Jun 25 2024 08:32:35 %S A330370 1,2,1,1,3,2,1,1,1,1,4,3,1,2,2,2,1,1,1,1,1,1,5,4,1,3,2,3,1,1,2,2,1,2, %T A330370 1,1,1,1,1,1,1,1,6,5,1,4,2,3,3,4,1,1,3,2,1,3,1,1,1,2,2,2,2,2,1,1,2,1, %U A330370 1,1,1,1,1,1,1,1,1,7,6,1,5,2,4,3,5,1,1,4,2,1,3,3,1,4,1,1,1,3,2,2,3,2,1,1,3 %N A330370 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = 1..n. %C A330370 Theorem: the k-th part of a partition in nonincreasing order of a positive integer equals the number of parts >= k of its conjugate partition. %C A330370 Example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition [3, 2, 2, 1, 1] 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. And vice versa, consider now the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition [5, 3, 1] 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. The fourth part is 1, so the conjugate partition has only one part >= 4. The fifth part is 1, so the conjugate partition has only one part >= 5. %C A330370 Corollary: the difference between the k-th part and the (k+1)-st part of a partition in nonincreasing order of a positive integer equals the number of k's in its conjugate partition. %C A330370 Example: consider the partition [5, 3, 1]. The difference between the first and the second parts is 5 - 3 = 2, which equals the number of 1's in its conjugate partition [3, 2, 2, 1, 1]. The difference between the second and third parts is 3 - 1 = 2, which equals the number of 2's in its conjugate partition. The difference between the third part and the fourth (virtual) part is 1 - 0 = 1, which equals the number of 3's in its conjugate partition. And vice versa, consider the partition [3, 2, 2, 1, 1]. The difference between the first and second parts is 3 - 2 = 1, which equals the number of 1's in its conjugate partition [5, 3, 1]. The difference between the second and third parts is 2 - 2 = 0, which equals the number of 2's in its conjugate partition. The difference between the third and fourth parts is 2 - 1 = 1, which equals the number of 3's in its conjugate partition, and so on. %C A330370 Self-conjugate partitions are included in all the above comments. %C A330370 A proof without words is as shown below: %C A330370 . %C A330370 +------------------------+ %C A330370 | +--------------------+ | %C A330370 | | +----------------+ | | %C A330370 | | | | | | %C A330370 v v v P2 FD k | | | %C A330370 | | | %C A330370 +--------> * * * 3 1 1 --+ | | %C A330370 | +------> * * 2 0 2 | | %C A330370 | +------> * * 2 1 3 ----+ | %C A330370 | | +----> * 1 0 4 | %C A330370 | | +----> * 1 1 5 ------+ %C A330370 | | | %C A330370 | | | P1 5 3 1 %C A330370 | | | %C A330370 | | | FD 2 2 1 %C A330370 | | | %C A330370 | | | k 1 2 3 %C A330370 | | | %C A330370 | | | | | | %C A330370 | | +-------+ | | %C A330370 | +-----------+ | %C A330370 +---------------+ %C A330370 . %C A330370 Every partition of n has n ranks. %C A330370 The k-th rank of a partition is the k-th part minus the number of parts >= k. %C A330370 In accordance with the above theorem, the k-th rank of a partition is also the number of parts >= k of its conjugate partition minus the number of parts >= k of the partition. %C A330370 All ranks of a partition are zeros if and only if the partition is a self-conjugate partition. %C A330370 The list of ranks of a partition of n equals the list of ranks multipled by -1 of its conjugate partition. %C A330370 For example, the nine ranks of the partition [5, 3, 1] 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 A330370 Note that the first rank coincides with the Dyson's rank because the first part of a partition is also the largest part, and the number of parts >= 1 is also the total number of parts. %C A330370 In this triangle the partitions of n appears ordered by their first rank. The partitions that have the same first rank appears ordered by their second rank. The partitions that have the same first rank and the same second rank appears ordered by their third rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on. %C A330370 Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n. %C A330370 The first ranks of the partitions of this triangle give A330368. %C A330370 For more information about the k-th ranks see A208478. %C A330370 First differs from A080577 at a(48), and from A036037 at a(56), and from A181317 at a(105). %e A330370 Triangle begins: %e A330370 [1]; %e A330370 [2], [1,1]; %e A330370 [3], [2,1], [1,1,1]; %e A330370 [4], [3,1], [2,2], [2,1,1], [1,1,1,1]; %e A330370 [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1]; %e A330370 [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2], ... %e A330370 ... %e A330370 Illustration of initial terms with a symmetric arrangement (note that the self-conjugate partitions are located in the main diagonal): %e A330370 . %e A330370 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A330370 * * * * * * * * * * * * * * * * * * * * * %e A330370 2 %e A330370 * %e A330370 * %e A330370 3 2 1 2 1 1 2 1 1 1 2 1 1 1 1 %e A330370 * * * * * * * * * * * * * * * %e A330370 * * * * * %e A330370 * %e A330370 4 3 1 2 2 2 2 1 2 2 1 1 %e A330370 * * * * * * * * * * * * %e A330370 * * * * * * * * %e A330370 * * %e A330370 * %e A330370 5 4 1 3 2 3 1 1 2 2 2 %e A330370 * * * * * * * * * * * %e A330370 * * * * * * * * %e A330370 * * * * %e A330370 * * 3 1 1 1 %e A330370 * * * * * %e A330370 * %e A330370 * %e A330370 . %e A330370 6 5 1 4 2 3 3 4 1 1 3 2 1 %e A330370 * * * * * * * * * * * * * %e A330370 * * * * * * * * * %e A330370 * * * * * * * %e A330370 * * * * %e A330370 * * %e A330370 * %e A330370 For n = 9 the 9th row of the triangle contains the partitions ordered as shown below: %e A330370 --------------------------------------------------------------------------------- %e A330370 Ranks %e A330370 Conjugate %e A330370 Label with Partitions k = 1 2 3 4 5 6 7 8 9 %e A330370 --------------------------------------------------------------------------------- %e A330370 1 30 [9] 8 -1 -1 -1 -1 -1 -1 -1 -1 %e A330370 2 29 [8, 1] 6 0 -1 -1 -1 -1 -1 -1 0 %e A330370 3 28 [7, 2] 5 0 -1 -1 -1 -1 -1 0 0 %e A330370 4 27 [6, 3] 4 1 -2 -1 -1 -1 0 0 0 %e A330370 5 26 [7, 1, 1] 4 0 0 -1 -1 -1 -1 0 0 %e A330370 6 25 [5, 4] 3 2 -2 -2 -1 0 0 0 0 %e A330370 7 24 [6, 2, 1] 3 0 0 -1 -1 -1 0 0 0 %e A330370 8 23 [5, 3, 1] 2 1 -1 -1 -1 0 0 0 0 %e A330370 9 22 [6, 1, 1, 1] 2 0 0 0 -1 -1 0 0 0 %e A330370 10 21 [5, 2, 2] 2 -1 1 -1 -1 0 0 0 0 %e A330370 11 20 [4, 4, 1] 1 2 -1 -2 0 0 0 0 0 %e A330370 12 19 [5, 2, 1, 1] 1 0 0 0 -1 0 0 0 0 %e A330370 13 18 [4, 3, 2] 1 0 0 -1 0 0 0 0 0 %e A330370 14 17 [4, 3, 1, 1] 0 1 -1 0 0 0 0 0 0 %e A330370 15 (self-conjugate) [5, 1, 1, 1, 1] All zeros -> 0 0 0 0 0 0 0 0 0 %e A330370 16 (self-conjugate) [3, 3, 3] All zeros -> 0 0 0 0 0 0 0 0 0 %e A330370 17 14 [4, 2, 2, 1] 0 -1 1 0 0 0 0 0 0 %e A330370 18 13 [3, 3, 2, 1] -1 0 0 1 0 0 0 0 0 %e A330370 19 12 [4, 2, 1, 1, 1] -1 0 0 0 1 0 0 0 0 %e A330370 20 11 [3, 2, 2, 2] -1 -2 1 2 0 0 0 0 0 %e A330370 21 10 [3, 3, 1, 1, 1] -2 1 -1 1 1 0 0 0 0 %e A330370 22 9 [4, 1, 1, 1, 1, 1] -2 0 0 0 1 1 0 0 0 %e A330370 23 8 [3, 2, 2, 1, 1] -2 -1 1 1 1 0 0 0 0 %e A330370 24 7 [3, 2, 1, 1, 1, 1] -3 0 0 1 1 1 0 0 0 %e A330370 25 6 [2, 2, 2, 2, 1] -3 -2 2 2 1 0 0 0 0 %e A330370 26 5 [3, 1, 1, 1, 1, 1, 1] -4 0 0 1 1 1 1 0 0 %e A330370 27 4 [2, 2, 2, 1, 1, 1] -4 -1 2 1 1 1 0 0 0 %e A330370 28 3 [2, 2, 1, 1, 1, 1, 1] -5 0 1 1 1 1 1 0 0 %e A330370 29 2 [2, 1, 1, 1, 1, 1, 1, 1] -6 0 1 1 1 1 1 1 0 %e A330370 30 1 [1, 1, 1, 1, 1, 1, 1, 1, 1] -8 1 1 1 1 1 1 1 1 %e A330370 . %e A330370 Two examples of the order of partitions: %e A330370 1) The partitions [6, 3] and [7, 1, 1] both have their first rank equal to 4, so they are ordered by their sencond rank. %e A330370 2) The self-conjugate partitions [5, 1, 1, 1, 1] and [3, 3, 3] both have all their ranks equal to zero, so they are ordered by their first part. %Y A330370 Row n contains A000041(n) partitions. %Y A330370 Row n has length A006128(n). %Y A330370 The sum of n-th row is A066186(n). %Y A330370 Cf. A000700, A036037, A067619, A080577, A181317, A207031, A330368, A330372, A330373. %Y A330370 For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483. %K A330370 nonn,tabf %O A330370 1,2 %A A330370 _Omar E. Pol_, Dec 12 2019