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A003293 Number of planar partitions of n decreasing across rows.
(Formerly M1058)
31
1, 1, 2, 4, 7, 12, 21, 34, 56, 90, 143, 223, 348, 532, 811, 1224, 1834, 2725, 4031, 5914, 8638, 12540, 18116, 26035, 37262, 53070, 75292, 106377, 149738, 209980, 293473, 408734, 567484, 785409, 1083817, 1491247, 2046233, 2800125, 3821959, 5203515 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of planar partitions monotonically decreasing down antidiagonals (i.e., with b(n,k) <= b(n-1,k+1)). Transpose (to get planar partitions decreasing down columns), then take the conjugate of each row. - Franklin T. Adams-Watters, May 15 2006

Also number of partitions into one kind of 1's and 2's, two kinds of 3's and 4's, three kinds of 5's and 6's, etc. - Joerg Arndt, May 01 2013

Also count of semistandard Young tableaux with sum of entries equal to n (row sums of A228125). - Wouter Meeussen, Aug 11 2013

REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 133.

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

LINKS

Nathaniel Johnston and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Nathaniel Johnston)

M. S. Cheema and W. E. Conway, Numerical investigation of certain asymptotic results in the theory of partitions, Math. Comp., 26 (1972), 999-1005.

B. Gordon and L. Houten, Notes on Plane Partitions I, J. of Comb. Theory, 4 (1968), 72-80.

B. Gordon and L. Houten, Notes on Plane Partitions II, J. of Comb. Theory, 4 (1968), 81-99.

Basil Gordon and Lorne Houten, Notes on plane partitions III, (first page is available), Duke Math. J. Volume 36, Number 4 (1969), 801-824.

B. Gordon and L. Houten, Notes on Plane Partitions V, Journal of Combinatorial Theory, vol. 11, issue 2, 1971, p. 157-168.

B. Gordon and L. Houten, Notes on Plane Partitions VI, Discrete Mathematics, vol. 26, issue 1, 1979, p. 41-45.

Vaclav Kotesovec, Graph - asymptotic ratio for 10000 terms

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

R. P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.

Richard P. Stanley, Theory and Application of Plane Partitions. Part 2

FORMULA

G.f.: Product_(1 - x^k)^{-c(k)}, c(k) = 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ....

Euler transform of A110654. - Michael Somos, Sep 19 2006

a(n) ~ 2^(-3/4) * (3*Pi*Zeta(3))^(-1/2) * (n/Zeta(3))^(-49/72) * exp(3/2*Zeta(3) * (n/Zeta(3))^(2/3) + Pi^2*(n/Zeta(3))^(1/3)/24 - Pi^4/(3456*Zeta(3)) + Zeta'(-1)/2) [Basil Gordon and Lorne Houten, 1969]. - Vaclav Kotesovec, Feb 28 2015

EXAMPLE

From Gus Wiseman, Jan 17 2019: (Start)

The a(6) = 21 plane partitions with strictly decreasing columns (the count is the same as for strictly decreasing rows):

  6   51   42   411   33   321   3111   222   2211   21111   111111

.

  5   4   41   31   32   311   22   221   2111

  1   2   1    2    1    1     11   1     1

.

  3

  2

  1

(End)

MAPLE

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(modp(n, 2)=0, n, n+1)/2): seq(a(n), n=0..45);  # Alois P. Heinz, Sep 08 2008

MATHEMATICA

CoefficientList[Series[Product[(1-x^k)^(-Ceiling[k/2]),

{k, 1, 40}], {x, 0, 40}], x][[1 ;; 40]]

(* Jean-Fran├žois Alcover, Apr 18 2011, after Michael Somos *)

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+1-(-1)^k)/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 28 2015 *)

nmax = 50; CoefficientList[Series[Product[1/((1-x^(2*k-1))*(1-x^(2*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2015 *)

PROG

(PARI) {a(n)=if(n<0, 0, polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n))} /* Michael Somos, Sep 19 2006 */

CROSSREFS

Cf. A005308, A005986.

Cf. A000085, A000219, A053529, A138178, A323432, A323436.

Sequence in context: A182746 A100482 A301762 * A192759 A289249 A094974

Adjacent sequences:  A003290 A003291 A003292 * A003294 A003295 A003296

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Feb 06 2000

Additional comments from Michael Somos, May 19 2000

STATUS

approved

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)