A259094 From the Lecture Hall Theorem: array read by antidiagonals: T(n,k) = number of partitions of n into odd parts of size < 2k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 1, 1, 1, 1, 2, 2, 3, 4, 3, 1, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 6, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 8, 5, 1

Offset

1,14

Comments

The Lecture Hall Theorem states that (the number of partitions (d1,d2,...,dn) of m such that 0 <= d1/1 <= d2/2 <= ... <= dn/n) equals (the number of partitions of m into odd parts less than 2n}.

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions, The Ramanujan J. 1 (1997) 101-111.

M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, The Ramanujan J. 1 (1997) 165-185.

Mireille Bousquet-Mélou, Kimmo Eriksson, A Refinement of the Lecture Hall Theorem, Journal of Combinatorial Theory, Series A, Volume 86, Issue 1, April 1999, Pages 63-84

Niklas Eriksen, A simple bijection between lecture hall partitions and partitions into odd integers Formal Power Series and Algebraic Combinatorics. 2002.

Robin Whitty, The Lecture Hall Partition Theorem

A. J. Yee, On combinatorics of lecture hall partitions, The Ramanujan J. 5 (2001) 247-262.

Example

The array begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ...
1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14 ...
1,1,1,2,2,3,4,4,5,6,7,8,9,10,11,13,14,15,17,18,20,22,23,25,27,29,31,33,35,37,40,42,44,47,49,52,55 ...
1,1,1,2,2,3,4,5,6,7,9,10,12,14,16,19,21,24,27,30,34,38,42,46,51,56,61,67,73,79,86,93,100,108,116 ...
1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58,65,74,82,91,102,113,124,138,151,165,182 ...
1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227 ...
1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,26,31,36,43,50,58,68,78,90,103,118,134,153,173,195,220,247,277 ...
1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,37,45,52,61,72,83,96,111,128,146,168,191,217,247,279,314 ...
...
The successive antidiagonals are:
[1]
[1, 1]
[1, 1, 1]
[1, 1, 1, 1]
[1, 1, 1, 2, 1]
[1, 1, 1, 2, 2, 1]
[1, 1, 1, 2, 2, 2, 1]
[1, 1, 1, 2, 2, 3, 3, 1]
[1, 1, 1, 2, 2, 3, 4, 3, 1]
[1, 1, 1, 2, 2, 3, 4, 4, 3, 1]
[1, 1, 1, 2, 2, 3, 4, 5, 5, 4, 1]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 6, 4, 1]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 4, 1]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 8, 5, 1]
...

Maple

G:=n->mul(1/(1-q^(2*i-1)), i=1..n);
M:=41;
G2:=n->seriestolist(series(G(n), q, M));
for n from 1 to 10 do lprint(G2(n)); od:
H:=n->[seq(G2(n-i+1)[i], i=1..n)];
for n from 1 to 14 do lprint(H(n)); od:

Mathematica

G[n_] := Product[1/(1-q^(2*i-1)), {i, 1, n}];
M = 41;
G2[n_] := CoefficientList[Series[G[n], {q, 0, M}], q];
For[n = 1, n <= 10, n++; Print[G2[n]]];
H[n_] := Table[G2[n-i+1][[i]], {i, 1, n}];
Reap[For[n = 1, n <= 14, n++, Print[H[n]]; Sow[H[n]]]][[2, 1]] // Flatten (* Jean-François Alcover, Jun 04 2017, translated from Maple *)

Crossrefs

Many rows of the array are already in the OEIS: A008620, A008672, A008673, A008674, A008675, A287997, A287998, A288000, A288001.

Sequence in context: A163100 A139038 A322812 * A306741 A274193 A238384

Adjacent sequences: A259091 A259092 A259093 * A259095 A259096 A259097

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 19 2015

Status

approved