# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a003293 Showing 1-1 of 1 %I A003293 M1058 #107 Oct 29 2023 01:44:05 %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, Table of n, a(n) for n = 0..10000 (first 1001 terms from Nathaniel Johnston) %H A003293 M. S. Cheema and W. E. Conway, Numerical investigation of certain asymptotic results in the theory of partitions, Math. Comp., 26 (1972), 999-1005. %H A003293 Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020. %H A003293 B. Gordon and L. Houten, Notes on Plane Partitions I, J. of Comb. Theory, 4 (1968), 72-80. %H A003293 B. Gordon and L. Houten, Notes on Plane Partitions II, J. of Comb. Theory, 4 (1968), 81-99. %H A003293 Basil Gordon and Lorne Houten, Notes on plane partitions III (first page is available), Duke Math. J. Volume 36, Number 4 (1969), 801-824. %H A003293 B. Gordon and L. Houten, Notes on Plane Partitions V, Journal of Combinatorial Theory, vol. 11, issue 2, 1971, pp. 157-168. %H A003293 B. Gordon and L. Houten, Notes on Plane Partitions VI, Discrete Mathematics, vol. 26, issue 1, 1979, pp. 41-45. %H A003293 Vaclav Kotesovec, Graph - asymptotic ratio for 10000 terms. %H A003293 Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016. %H A003293 Richard P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279. %H A003293 Richard P. Stanley, Theory and Application of Plane Partitions. Part 2, 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 A. Sellers_, Feb 06 2000 %E A003293 Additional comments from _Michael Somos_, May 19 2000 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE