|
|
A114088
|
|
Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).
|
|
39
|
|
|
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 5, 6, 6, 5, 3, 2, 1, 1, 1, 6, 8, 8, 7, 5, 3, 2, 1, 1, 1, 7, 10, 10, 9, 7, 5, 3, 2, 1, 1, 1, 9, 13, 13, 12, 10, 7, 5, 3, 2, 1, 1, 1, 10, 16, 17, 15, 13, 10, 7, 5, 3, 2, 1, 1, 1, 12, 20, 22, 20, 17
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,7
|
|
COMMENTS
|
Also the number of integer partitions of n with k parts below the diagonal. For example, the partition (3,2,2,1) has two parts (at positions 3 and 4) below the diagonal (1,2,3,4). Row n = 8 counts the following partitions:
8 71 611 5111 41111 311111 2111111 11111111
44 332 2222 22211 221111
53 422 3221 32111
62 431 3311
521 4211
Indices of parts below the diagonal are also called strong nonexcedances.
(End)
|
|
REFERENCES
|
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
|
|
LINKS
|
|
|
FORMULA
|
G.f. = Sum_{k>=1} q^(k^2) / Product_{j=1..k} (1 - q^j)*(1 - t*q^j).
Sum_{k=0..n-1} k*T(n,k) = A114089(n).
|
|
EXAMPLE
|
T(7,2)=3 because we have [5,1,1], [3,2,1,1] and [2,2,2,1] (the bottom tails are [1,1], [1,1] and [2,1], respectively).
Triangle starts:
1;
1, 1;
1, 1, 1;
2, 1, 1, 1;
2, 2, 1, 1, 1;
3, 3, 2, 1, 1, 1;
3, 4, 3, 2, 1, 1, 1;
|
|
MAPLE
|
g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j), j=1..k), k=1..20): gserz:=simplify(series(g, z=0, 30)): for n from 1 to 14 do P[n]:=coeff(gserz, z^n) od: for n from 1 to 14 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form
|
|
MATHEMATICA
|
subdiags[y_]:=Length[Select[Range[Length[y]], #>y[[#]]&]];
Table[Length[Select[IntegerPartitions[n], subdiags[#]==k&]], {n, 1, 15}, {k, 0, n-1}] (* Gus Wiseman, May 21 2022 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|