A138178 -id:A138178 - cp's OEIS frontend

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120

Offset: 1

Gus Wiseman, Feb 14 2018

nonn

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
ssyt[n_]:=If[n===1,1,Sum[ssyt[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Table[ssyt[conj[n]],{n,50}]

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 11, 12, 16, 12, 32, 28, 31, 8, 64, 31, 128, 33, 82, 64, 256, 28, 69, 144, 69, 86, 512, 105, 1024, 16, 208, 320, 209, 82, 2048, 704, 512, 86, 4096, 318, 8192, 216, 262, 1536, 16384, 64, 465, 262, 1232, 528, 32768, 209, 588, 245, 2912, 3328

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 11 tableaux:
1 1
1 1
.
2 1   1 1   1 1   1 2
1 1   1 2   2 2   1 2
.
1 1   1 2   1 2   1 3
2 3   1 3   3 3   2 3
.
1 2   1 3
3 4   2 4

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]Table[PrimePi[p],{k}]]]];
Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]

A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 17, 28, 41, 75, 122, 192, 314, 484, 771, 1216, 1861, 2848, 4395, 6610, 10037

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Jan 17 2019: (Start)
Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
  6   51   42   321
.
  5   4   41   31   32   31   22   221   211
  1   2   1    2    1    11   2    1     11
.
  3   21   21   111
  2   2    11   11
  1   1    1    1
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{2},{1,2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{1,1},{1,1,1}}
   {{1,2},{1,2,2}}
   {{2,2},{1,2,2}}

Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A000085, A138178, A323436.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* Gus Wiseman, Jan 18 2019 *)

a(11)-a(17) from Gus Wiseman, Jan 18 2019

a(18)-a(21) from Robert Price, Jun 21 2021

A003293 Number of planar partitions of n decreasing across rows.

Original entry on oeis.org

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

A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 3, 15, 107, 991, 11267, 151721, 2360375, 41650861, 821881709, 17932031225, 428630422697, 11138928977049, 312680873171465, 9428701154866535, 303957777464447449, 10431949496859168189, 379755239311735494421

Offset: 0

Vladeta Jovovic, Aug 18 2006

nonn

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 15 matrices:
  [3]
.
  [2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k,m]^2,{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k).
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n.
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - Paul D. Hanna, Mar 26 2018

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A104602 Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.

Original entry on oeis.org

1, 1, 2, 10, 70, 642, 7246, 97052, 1503700, 26448872, 520556146, 11333475922, 270422904986, 7016943483450, 196717253145470, 5925211960335162, 190825629733950454, 6543503207678564364, 238019066600097607402, 9153956822981328930170, 371126108428565106918404

Offset: 0

Ralf Stephan, Mar 27 2005

nonn

Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns, up to row and column permutation, is A057151(n). - Vladeta Jovovic, Mar 25 2006

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 10 matrices:
  [1 1] [1 1] [1 0] [0 1]
  [1 0] [0 1] [1 1] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Row sums of triangle A104601.

Cf. A048291, A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}],{k,0,n}],{n,1,20}] (* Vaclav Kotesovec, May 07 2014 *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - Vladeta Jovovic, Mar 25 2006
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n*(n+1)). - Paul D. Hanna, Mar 26 2018

More terms from Vladeta Jovovic, Mar 25 2006

a(0)=1 prepended by Alois P. Heinz, Jan 14 2015

A296561 Number of rim-hook (or border-strip) tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 10, 12, 16, 12, 32, 28, 29, 8, 64, 29, 128, 33, 78, 64, 256, 28, 62, 144, 62, 86, 512, 100, 1024, 16, 200, 320, 193, 78, 2048, 704, 496, 86, 4096, 306, 8192, 216, 242, 1536, 16384, 64, 414, 242, 1200, 528, 32768, 193, 552, 245, 2848, 3328

Offset: 1

Gus Wiseman, Feb 15 2018

nonn

The Murnaghan-Nakayama rule uses rim-hook tableaux to expand Schur functions in terms of power-sum symmetric functions.

			The a(6) = 5 tableaux:
3 2   3 1   2 2   2 1   1 1
1     2     1     2     1

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.17.

Wikipedia, Murnaghan-Nakayama rule

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A296150, A296188, A299202.

A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296

Offset: 0

Gus Wiseman, Feb 26 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3   1 2 4   1 3 4   1 2 3   1 2 3
4       3       2       2       3
.
1 2   1 3   1 2
3 4   2 4   2 3
.
1 2   1 3   1 2   1 4   1 3
3     2     2     2     2
4     4     3     3     3
.
1
2
3
4

Alois P. Heinz, Table of n, a(n) for n = 0..50 (first 46 terms from Ludovic Schwob)
D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics, Vol. 34, No. 3 (1970), 709-727.
Wikipedia, Young tableau

Cf. A000085, A000898, A001221, A005117, A006958, A015128, A138178, A238121, A238690, A285175, A296561, A297388, A299926, A300120, A300122.

unddis[y_]:=DeleteCases[y-#,0]&/@Tuples[Table[If[y[[i]]>Append[y,0][[i+1]],{0,1},{0}],{i,Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]},If[Length[sam]===0,If[Total[y]===0,{{}},{}],Join@@Table[Prepend[#,y]&/@dos[sam[[k]]],{k,1,Length[sam]}]]];
Table[Sum[Length[dos[y]],{y,IntegerPartitions[n]}],{n,1,8}]

a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023

More terms from Ludovic Schwob, Sep 23 2023

cp's OEIS Frontend

A000898 a(n) = 2*(a(n-1) + (n-1)*a(n-2)) for n >= 2 with a(0) = 1.

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A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

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A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

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A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

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A003293 Number of planar partitions of n decreasing across rows.

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A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

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A321405 Number of non-isomorphic self-dual set systems of weight n.

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A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.