cp's OEIS Frontend

A003293 Number of planar partitions of n decreasing across rows.

%I A003293 M1058 #108 Jul 02 2025 16:01:54
%S A003293 1,1,2,4,7,12,21,34,56,90,143,223,348,532,811,1224,1834,2725,4031,
%T A003293 5914,8638,12540,18116,26035,37262,53070,75292,106377,149738,209980,
%U A003293 293473,408734,567484,785409,1083817,1491247,2046233,2800125,3821959,5203515
%N A003293 Number of planar partitions of n decreasing across rows.
%C A003293 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 A003293 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 A003293 Also count of semistandard Young tableaux with sum of entries equal to n (row sums of A228125). - _Wouter Meeussen_, Aug 11 2013
%D A003293 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 133.
%D A003293 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003293 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 A003293 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 A003293 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 A003293 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 A003293 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 A003293 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 A003293 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 A003293 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 A003293 Vaclav Kotesovec, <a href="/A003293/a003293.jpg">Graph - asymptotic ratio for 10000 terms</a>.
%H A003293 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 A003293 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 A003293 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 A003293 G.f.: Product_(1 - x^k)^{-c(k)}, c(k) = 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ....
%F A003293 Euler transform of A110654. - _Michael Somos_, Sep 19 2006
%F A003293 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 A003293 From _Gus Wiseman_, Jan 17 2019: (Start)
%e A003293 The a(6) = 21 plane partitions with strictly decreasing columns (the count is the same as for strictly decreasing rows):
%e A003293   6   51   42   411   33   321   3111   222   2211   21111   111111
%e A003293 .
%e A003293   5   4   41   31   32   311   22   221   2111
%e A003293   1   2   1    2    1    1     11   1     1
%e A003293 .
%e A003293   3
%e A003293   2
%e A003293   1
%e A003293 (End)
%p A003293 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 A003293 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 A003293 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 A003293 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 A003293 (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 A003293 Cf. A005308, A005986.
%Y A003293 Cf. A000085, A000219, A053529, A138178, A323432, A323436.
%K A003293 nonn,easy,nice
%O A003293 0,3
%A A003293 _N. J. A. Sloane_
%E A003293 More terms from _James Sellers_, Feb 06 2000
%E A003293 Additional comments from _Michael Somos_, May 19 2000