A005986 -id:A005986 - cp's OEIS frontend

A003293 Number of planar partitions of n decreasing across rows.

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

Offset: 0

N. J. A. Sloane

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

			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)

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).

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.
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.
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, pp. 157-168.
B. Gordon and L. Houten, Notes on Plane Partitions VI, Discrete Mathematics, vol. 26, issue 1, 1979, pp. 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], 2015-2016.
Richard 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, Studies in Appl. Math., 1 (1971), 259-279.

Cf. A005308, A005986.

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

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

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 *)
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 *)

{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 */

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

More terms from James Sellers, Feb 06 2000

Additional comments from Michael Somos, May 19 2000

A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 5, 2, 1, 1, 6, 10, 9, 5, 2, 1, 1, 7, 14, 16, 10, 5, 2, 1, 1, 8, 19, 24, 19, 11, 5, 2, 1, 1, 9, 24, 37, 32, 21, 11, 5, 2, 1, 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1, 1, 11, 37, 71, 79, 66, 41, 23, 11, 5, 2, 1, 1, 12, 44, 93, 117, 106, 74, 43, 23, 11, 5, 2, 1, 1, 13, 52, 122, 166, 166, 125, 80, 44, 23, 11, 5, 2, 1, 1, 14, 61, 153, 231, 251, 204, 139, 83, 45, 23, 11, 5, 2, 1, 1, 15, 70, 193, 311, 367, 322, 236, 147, 85, 45, 23, 11, 5, 2, 1

Offset: 1

Wouter Meeussen, Aug 11 2013

Row sums equal A003293.

Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ...

			T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are
114 , 123 , 222 , 11 ,  12  , 13 ,   1
                  4     3     2      2
                                     3
Triangle starts:
1;
1,  1;
1,  2,  1;
1,  3,  2,  1;
1,  4,  4,  2,  1;
1,  5,  7,  5,  2,  1;
1,  6, 10,  9,  5,  2,  1;
1,  7, 14, 16, 10,  5,  2,  1;
1,  8, 19, 24, 19, 11,  5,  2, 1;
1,  9, 24, 37, 32, 21, 11,  5, 2, 1;
1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;

Cf. A003293, A005986.

hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q>=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];
Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1&) /@ Flatten[hooklength[#1]],{q,0,w}]&) /@ Partitions[n]],{w,24},{n,w}]

A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

Original entry on oeis.org

1, 1, 4, 7, 19, 35, 80, 149, 307, 566, 1092, 1974, 3643, 6447, 11498, 19947, 34636, 58974, 100182, 167713, 279659, 461056, 756562, 1230104, 1990255, 3195471, 5105540, 8103722, 12801925, 20107448, 31439978, 48907179, 75755094, 116797754, 179354540, 274253042

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only one or two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:
  {{1}}  {{11}}    {{111}}      {{1111}}
         {{12}}    {{122}}      {{1122}}
         {{1}{1}}  {{1}{11}}    {{1222}}
         {{1}{2}}  {{1}{22}}    {{1}{111}}
                   {{2}{12}}    {{11}{11}}
                   {{1}{1}{1}}  {{1}{122}}
                   {{1}{2}{2}}  {{11}{22}}
                                {{12}{12}}
                                {{1}{222}}
                                {{12}{22}}
                                {{2}{122}}
                                {{1}{1}{11}}
                                {{1}{1}{22}}
                                {{1}{2}{12}}
                                {{1}{2}{22}}
                                {{2}{2}{12}}
                                {{1}{1}{1}{1}}
                                {{1}{1}{2}{2}}
                                {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:
  {{1}}  {{11}}    {{111}}    {{1111}}
         {{12}}    {{122}}    {{1122}}
         {{1}{1}}  {{123}}    {{1222}}
         {{1}{2}}  {{1}{11}}  {{1233}}
                   {{1}{22}}  {{1234}}
                   {{1}{23}}  {{1}{111}}
                   {{2}{12}}  {{11}{11}}
                              {{1}{122}}
                              {{11}{22}}
                              {{12}{12}}
                              {{1}{222}}
                              {{12}{22}}
                              {{1}{233}}
                              {{12}{33}}
                              {{1}{234}}
                              {{12}{34}}
                              {{13}{23}}
                              {{2}{122}}
                              {{3}{123}}
Inequivalent representatives of the a(4) = 19 matrices:
  [4] [2 2] [1 3]
.
  [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]
  [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]
.
  [1] [1 0] [1 0] [1 0] [0 1]
  [1] [1 0] [0 1] [0 1] [0 1]
  [2] [0 2] [1 1] [0 2] [1 1]
.
  [1] [1 0] [1 0]
  [1] [1 0] [0 1]
  [1] [0 1] [0 1]
  [1] [0 1] [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A007716, A052847, A005380, A054974, A005986, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A005308 Bosonic string states.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 8, 14, 16, 25, 31, 47, 58, 85, 107, 153, 195, 271, 348, 480, 616, 834, 1077, 1445, 1863, 2478, 3194, 4216, 5431, 7118, 9157, 11942, 15329, 19884, 25485, 32916, 42090, 54147, 69093, 88563, 112769, 144056, 183028, 233112, 295525

Offset: 1

N. J. A. Sloane

nonn

See the reference for precise definition.

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
T. Curtright, Counting symmetry patterns in the spectra of strings, in H. J. de Vega and N. Sánchez, editors, String Theory, Quantum Cosmology and Quantum Gravity. Integrable and Conformal Invariant Theories. World Scientific, Singapore, 1987, pp. 304-333.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Cf. A003293, A005986.

nmax = 50; Rest[CoefficientList[Series[x/(1-x)*Product[1/(1-x^k)^((2*k - 5 + (-1)^k)/4), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2016 *)

G.f.: Product (1 - x^k)^{-c(k)}; c(k) = 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ....
Euler transform gives sequence with g.f. = x^3/((x+1)*(x-1)^2), Simon Plouffe, Master's Thesis, UQAM 1992.
a(n) ~ 2^(1/4) * exp(1/24 - 25*Pi^4/(3456*Zeta(3)) - 5*Pi^2 * n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2) / (A^(1/2) * sqrt(3) * Zeta(3)^(23/72) * n^(13/72)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 26 2016

A228128 T(n,m) = semistandard Young tableau families, headed by a father SSYT with shape a partition of k, containing daughter SSYT of shape equal to once-trimmed father's shape, so that union of families equals all SSYT with sum of entries n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 3, 4, 3, 1, 1, 0, 1, 4, 7, 5, 3, 1, 1, 0, 1, 5, 8, 9, 6, 3, 1, 1, 0, 1, 5, 13, 13, 10, 6, 3, 1, 1, 0, 1, 6, 14, 20, 17, 11, 6, 3, 1, 1, 0, 1, 7, 20, 27, 28, 19, 12, 6, 3, 1, 1, 0, 1, 7, 22, 38, 40, 33, 20, 12, 6, 3, 1, 1, 0, 1, 8, 29, 49, 60, 51, 37, 21, 12, 6, 3, 1, 1, 0, 1, 9, 31, 65, 85, 79, 59, 39, 22, 12, 6, 3, 1, 1

Offset: 1

Wouter Meeussen, Aug 11 2013

Row sums are A228129.

Reverse of rows seem to converge to first differences of A005986.

			T(6,3) = 3 since the 7 tableaux in the family contain 3 father tableaux:
11  ,  13  ,  1
4      2      2
              3
see 2nd link, "content 6".

N. Dragon, R. Stanley, Semi-Standard Young Diagrams and families;
N. Dragon, résumé

Cf. A228125, A228128, A228129.

(* hooklength: see A228125 *);
Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1 &) /@ Flatten[hooklength[#1]],{q,0,w}]& ) /@ Partitions[n]],{w,24},{n,w}]

cp's OEIS Frontend

A003293 Number of planar partitions of n decreasing across rows.

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A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.

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A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

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A005308 Bosonic string states.

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A228128 T(n,m) = semistandard Young tableau families, headed by a father SSYT with shape a partition of k, containing daughter SSYT of shape equal to once-trimmed father's shape, so that union of families equals all SSYT with sum of entries n.

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