%I M1058 #107 Oct 29 2023 01:44:05

%S 1,1,2,4,7,12,21,34,56,90,143,223,348,532,811,1224,1834,2725,4031,

%T 5914,8638,12540,18116,26035,37262,53070,75292,106377,149738,209980,

%U 293473,408734,567484,785409,1083817,1491247,2046233,2800125,3821959,5203515

%N Number of planar partitions of n decreasing across rows.

%C 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

%C 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

%C Also count of semistandard Young tableaux with sum of entries equal to n (row sums of A228125). - _Wouter Meeussen_, Aug 11 2013

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

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

%H Alois P. Heinz, <a href="/A003293/b003293.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Nathaniel Johnston)

%H M. S. Cheema and W. E. Conway, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0314756-0">Numerical investigation of certain asymptotic results in the theory of partitions</a>, Math. Comp., 26 (1972), 999-1005.

%H Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, <a href="https://arxiv.org/abs/2004.08901">Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions</a>, arXiv:2004.08901 [math.CO], 2020.

%H B. Gordon and L. Houten, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80088-2">Notes on Plane Partitions I</a>, J. of Comb. Theory, 4 (1968), 72-80.

%H B. Gordon and L. Houten, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80089-4">Notes on Plane Partitions II</a>, J. of Comb. Theory, 4 (1968), 81-99.

%H Basil Gordon and Lorne Houten, <a href="http://dx.doi.org/10.1215/S0012-7094-69-03696-5">Notes on plane partitions III</a> (first page is available), Duke Math. J. Volume 36, Number 4 (1969), 801-824.

%H B. Gordon and L. Houten, <a href="http://dx.doi.org/10.1016/0095-8956(71)90026-8">Notes on Plane Partitions V</a>, Journal of Combinatorial Theory, vol. 11, issue 2, 1971, pp. 157-168.

%H B. Gordon and L. Houten, <a href="http://dx.doi.org/10.1016/0012-365X(79)90059-1">Notes on Plane Partitions VI</a>, Discrete Mathematics, vol. 26, issue 1, 1979, pp. 41-45.

%H Vaclav Kotesovec, <a href="/A003293/a003293.jpg">Graph - asymptotic ratio for 10000 terms</a>.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

%H Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan//pubs/pubfiles/12-2.pdf">Theory and Applications of Plane Partitions: Part 2</a>, Studies in Appl. Math., 1 (1971), 259-279.

%H Richard P. Stanley, <a href="https://doi.org/10.1002/sapm1971503259">Theory and Application of Plane Partitions. Part 2</a>, Studies in Appl. Math., 1 (1971), 259-279.

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

%F Euler transform of A110654. - _Michael Somos_, Sep 19 2006

%F 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

%e From _Gus Wiseman_, Jan 17 2019: (Start)

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

%e 6 51 42 411 33 321 3111 222 2211 21111 111111

%e .

%e 5 4 41 31 32 311 22 221 2111

%e 1 2 1 2 1 1 11 1 1

%e .

%e 3

%e 2

%e 1

%e (End)

%p 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

%t CoefficientList[Series[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, 40}], {x, 0, 40}], x][[1 ;; 40]] (* _Jean-François Alcover_, Apr 18 2011, after _Michael Somos_ *)

%t 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 *)

%t 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 *)

%o (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 */

%Y Cf. A005308, A005986.

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

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Feb 06 2000

%E Additional comments from _Michael Somos_, May 19 2000