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A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.

%I A105806 #35 Feb 16 2025 08:32:57
%S A105806 1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,2,1,1,0,1,3,1,2,1,1,0,1,2,3,2,2,1,1,
%T A105806 0,1,4,3,3,2,2,1,1,0,1,4,5,3,4,2,2,1,1,0,1,6,5,6,3,4,2,2,1,1,0,1,7,8,
%U A105806 6,6,4,4,2,2,1,1,0,1,11,8,9,7,6,4,4,2,2,1,1,0,1,11,13,10,10,7,7,4,4,2,2,1,1,0,1
%N A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.
%C A105806 The array with all ranks (including negative ones) is A063995.
%C A105806 a(n,-r)=a(n,r) for negative rank -r with r from 1,2,...,n-1 (due to conjugation of partitions of n; see the link).
%C A105806 Dyson's rank of a partition of n is the maximal part minus the number of parts, i.e. the number of columns minus the number of rows of the Ferrers diagram (see the link) of the partition.
%H A105806 Lars Blomberg, <a href="/A105806/b105806.txt">Table of n, a(n) for n = 1..5050</a>
%H A105806 Wolfdieter Lang, <a href="/A105806/a105806.txt">First 16 rows.</a>
%H A105806 Alexander Berkovich and Frank G. Garvan, <a href="https://doi.org/10.1006/jcta.2002.3281">Some observations on Dyson's new symmetries of partitions</a>, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.
%H A105806 Freeman J. Dyson, <a href="https://www.jstor.org/stable/2304400">Problems for solution nr. 4261</a>, Am. Math. Month. 54 (1947) 418.
%H A105806 Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 1.
%H A105806 Freeman J. Dyson, <a href="https://doi.org/10.1016/0097-3165(89)90043-5">Mappings and symmetries of partitions</a>, J. Combin. Theory Ser. A 51 (1989), 169-180.
%H A105806 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConjugatePartition.html">Conjugation of partitions of n.</a>
%H A105806 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers diagram.</a>
%F A105806 a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.
%F A105806 G.f. of column r: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(r*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ). - _Seiichi Manyama_, May 21 2023
%e A105806 Triangle starts:
%e A105806   1;
%e A105806   0, 1;
%e A105806   1, 0, 1;
%e A105806   1, 1, 0, 1;
%e A105806   1, 1, 1, 0, 1;
%e A105806   1, 2, 1, 1, 0, 1; ...
%e A105806 Row 6, second entry is 2 because there are 2 partitions of n=6 with rank r=2-1=1, namely (3^2) and (1^2,4).
%e A105806 The table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
%e A105806 n\m -6 -5  -4  -3  -2  -1   0   1   2   3   4   5   6
%e A105806 -----------------------------------------------------
%e A105806 0   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
%e A105806 1   0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,
%e A105806 2   0,  0,  0,  0,  0,  1,  0,  1,  0,  0,  0,  0,  0,
%e A105806 3   0,  0,  0,  0,  1,  0,  1,  0,  1,  0,  0,  0,  0,
%e A105806 4   0,  0,  0,  1,  0,  1,  1,  1,  0,  1,  0,  0,  0,
%e A105806 5   0,  0,  1,  0,  1,  1,  1,  1,  1,  0,  1,  0,  0,
%e A105806 6   0,  1,  0,  1,  1,  2,  1,  2,  1,  1,  0,  1,  0,
%e A105806 7   1,  0,  1,  1,  2,  1,  3,  1,  2,  1,  1,  0,  1,
%e A105806 ...
%e A105806 The central triangle is A063995, the right-hand triangle is the present sequence. - _N. J. A. Sloane_, Jan 23 2020
%Y A105806 For the full triangle see A063995.
%Y A105806 Columns for r=0..5 are given in A047993, A101198, A101199, A101200, A363213, A363214.
%Y A105806 Row sums = A064174.
%K A105806 nonn,easy,tabl
%O A105806 1,17
%A A105806 _Wolfdieter Lang_, Mar 11 2005