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Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.
16

%I #34 May 21 2023 10:26:39

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

%C The array with all ranks (including negative ones) is A063995.

%C 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 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 Lars Blomberg, <a href="/A105806/b105806.txt">Table of n, a(n) for n = 1..5050</a>

%H Wolfdieter Lang, <a href="/A105806/a105806.txt">First 16 rows.</a>

%H 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 Freeman J. Dyson, <a href="https://www.jstor.org/stable/2304400">Problems for solution nr. 4261</a>, Am. Math. Month. 54 (1947) 418.

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConjugatePartition.html">Conjugation of partitions of n.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers diagram.</a>

%F a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.

%F 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 Triangle starts:

%e 1;

%e 0, 1;

%e 1, 0, 1;

%e 1, 1, 0, 1;

%e 1, 1, 1, 0, 1;

%e 1, 2, 1, 1, 0, 1; ...

%e 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 The table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):

%e n\m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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

%e 0 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

%e 1 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

%e 2 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0,

%e 3 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0,

%e 4 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0,

%e 5 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0,

%e 6 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 0,

%e 7 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1,

%e ...

%e The central triangle is A063995, the right-hand triangle is the present sequence. - _N. J. A. Sloane_, Jan 23 2020

%Y For the full triangle see A063995.

%Y Columns for r=0..5 are given in A047993, A101198, A101199, A101200, A363213, A363214.

%Y Row sums = A064174.

%K nonn,easy,tabl

%O 1,17

%A _Wolfdieter Lang_, Mar 11 2005