A006330 Number of corners, or planar partitions of n with only one row and one column. (Formerly M2553)

1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132

Offset

0,3

Comments

The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13 and here we have 12.

a(n) + A001523(n) = A000712(n). - Michael Somos, Jul 22 2003

Number of unimodal compositions of n+1 where the maximal part appears once, see example. [Joerg Arndt, Jun 11 2013]

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.

Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.

G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.

G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)

G. Kreweras, Letter to N. J. A. Sloane

Formula

G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.

G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - Michael Somos, Jul 28 2003

Convolution product of A197870 and A000712. - Michael Somos, Feb 22 2015

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

Example

From Joerg Arndt, Jun 11 2013: (Start)
There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once:
01:  [ 1 1 1 2 ]
02:  [ 1 1 2 1 ]
03:  [ 1 1 3 ]
04:  [ 1 2 1 1 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 1 1 ]
08:  [ 2 3 ]
09:  [ 3 1 1 ]
10:  [ 3 2 ]
11:  [ 4 1 ]
12:  [ 5 ]
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 21*x^5 + 38*x^6 + 63*x^7 + 106*x^8 + ...

Mathematica

a[0] = 1; a[n_] := SeriesCoefficient[ Sum[x^k/Product[1 - x^i, {i, 1, k}]^2, {k, 1, n}] + 1, {x, 0, n}]; Array[a, 39, 0] (* Jean-François Alcover, Mar 13 2014 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / prod(i=1, k, 1 - x^i, 1 + x*O(x^n))^2, 1), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) - 1)\2, (-1)^k * x^((k + k^2)/2)) / eta(x + x*O(x^n))^2, n))};

Crossrefs

Cf. A000219, A000712, A197870.

Column k=1 of A247255.

Row sums of A259100.

Sequence in context: A247662 A337462 A215005 * A293636 A087503 A092176

Adjacent sequences: A006327 A006328 A006329 * A006331 A006332 A006333

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited and extended by Moshe Shmuel Newman, Jun 10 2003

Status

approved