A006958 -id:A006958 - cp's OEIS frontend

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.

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}]

A259478 Partition containment triangle.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 8, 7, 5, 7, 12, 13, 12, 7, 11, 20, 23, 25, 19, 11, 15, 28, 35, 42, 39, 30, 15, 22, 42, 54, 70, 70, 66, 45, 22, 30, 58, 78, 105, 114, 119, 99, 67, 30, 42, 82, 112, 158, 178, 202, 186, 155, 97, 42, 56, 110, 154, 223, 262, 313, 314, 292, 226, 139, 56, 77, 152, 215, 319, 383, 479, 503, 511, 442, 336, 195, 77

Offset: 1

Wouter Meeussen, Jun 28 2015

T(n,k) counts pairs of partitions (lambda,mu) with Ferrers diagram of mu not extending beyond the diagram of lambda for all partitions lambda of size n and mu of size k <= n.
First column and main diagonal both equal A000041 (partition numbers).
This sequence counts (2,1)/(1) as different from (3,2,1)/(3,1) while their set-theoretic difference lambda - mu (their skew diagram) is the same.

			T(3,2) = 4, the pairs of partitions are ((3)/(2)), ((2,1)/(2)), ((2,1)/(1,1)), ((1,1,1)/(1,1))
and the diagrams are:
  x x 0 , x x , x 0 , x
          0     x     x
                      0
Triangle begins:
  n=1;  1
  n=2;  2  2
  n=3;  3  4  3
  n=4;  5  8  7  5
  n=5;  7 12 13 12  7
  n=6; 11 20 23 25 19 11

I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Columns k=1-10 give: A000041, 2*A000065, A303853, A303854, A303855, A303856, A303857, A303858, A303859, A303860.

Cf. A000070, A259479, A259480, A259481, A161492, A227309, A006958, A297388, A303851, A303852, A303861, A303862, A303863.

b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
      `if`(t=0, 1, add(x^j, j=0..n)), b(n, i-1, min(i-1, t))+
       add(b(n-i, min(i, n-i), min(j, n-i))*x^j, j=0..t)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$3)):
seq(T(n), n=1..15);  # Alois P. Heinz, Jul 05 2015, revised Jan 10 2018

majorsweak[left_List,right_List]:=Block[{le1=Length[left],le2=Length[right]},If[le2>le1||Min[Sign[left-PadRight[right,le1]]]<0,False,True]];
Table[Sum[ If[! majorsweak[\[Lambda], \[Mu]], 0, 1] , {\[Lambda], IntegerPartitions[n] }, {\[Mu], IntegerPartitions[m]}], {n, 7}, {m, n}]
(* Second program: *)
b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[m > n, 0, If[n == 0, 1, If[i < 1, 0, If[t && j > 0, b[n, m, i, j - 1, t], 0] + If[i > j, b[n, m, i - 1, j, False], 0] + If[i > n || j > m, 0, b[n - i, m - j, i, j, True]]]]]; T[n_, m_] :=  b[n, m, n, m, True]; Table[Table[T[n, m], {m, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Aug 27 2016, after Alois P. Heinz *)

Sum_{k=1..n} T(n,k) = A297388(n) - A000041(n). - Alois P. Heinz, Jan 10 2018

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

A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 213, 391, 723, 1333, 2463, 4547, 8403, 15522, 28686, 53006, 97963, 181042, 334606, 618415, 1142994, 2112545, 3904592, 7216810, 13338856, 24654268, 45568784, 84225393, 155675230, 287737327, 531830605, 982993368, 1816887637, 3358192905

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Sums along falling diagonals of A161492 (skew Ferrers diagrams by area and number of columns). [Joerg Arndt, Mar 23 2014]

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. P. Delest, J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics. vol.112, no.1-3, pp.65-79, (1993)

Cf. A049346 (g.f.: 1-1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[2, nmax] - Floor[Range[2, nmax]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+2) / G(k+1) ) );
Vec( 1 / G(0) )

/* formula from the Delest/Fedou reference with t=q: */
N=66;  q='q+O('q^N);  t=q;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
v=Vec(nm/dn)

G.f.: 1/(1-q /(1-q^2/(1-q^2/(1-q^3/(1-q^3/(1-q^4/(1-q^4/(1-q^5/(1-q^5/(1-...) )) )) )) )) ).
G.f.: 1/x - Q(0)/(2*x), where Q(k)= 1 + 1/(1 - 1/(1 - 1/(2*x^(k+1)) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: 1/x - U(0)/x, where U(k)= 1 - x^(k+1)/(1 - x^(k+1)/U(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: -W(0)/x, where W(k)= 1 - x^(k+1) - x^k - x^(2*k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: G(0) where G(k) = 1 - q/(q^(k+2) - 1 / G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2016
a(n) ~ c * d^n, where d = 1.84832326133106924642685135202616091890310896530577301386219207630312784... and c = 0.244648950328338656997216931963422920467577616734159139510762093105072... - Vaclav Kotesovec, Sep 05 2017

A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 7, 8, 11, 13, 16, 20, 24, 31, 37, 46, 58, 70, 88, 108, 133, 167, 204, 252, 315, 386, 479, 594, 731, 909, 1122, 1386, 1720, 2124, 2628, 3254, 4022, 4980, 6160, 7618, 9432, 11665, 14433, 17860, 22093, 27341, 33824, 41847, 51785, 64065, 79267

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Number of rough sandpiles: 1-dimensional sandpiles (see A186085) with n grains without flat steps (no two successive parts of the corresponding composition equal), see example. - Joerg Arndt, Mar 08 2014
The sequence of such sandpiles by base length starts (n>=0) 1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, ... (A097331, essentially A000108 with interlaced zeros).  This is a consequence of the obvious connection to Dyck paths, see example. - Joerg Arndt, Mar 09 2014
a(n>=1) are the Dyck paths with area n between the x-axis and the path which return to the x-axis only once (at their end), whereas A143951 includes paths with intercalated touches of the x-axis. - R. J. Mathar, Aug 22 2018

			From _Joerg Arndt_, Mar 08 2014: (Start)
The a(21) = 7 rough sandpiles are:
:
:   1:      [ 1 2 1 2 1 2 1 2 1 2 3 2 1 ]  (composition)
:
:           o
:  o o o o ooo
: ooooooooooooo  (rendering of sandpile)
:
:
:   2:      [ 1 2 1 2 1 2 1 2 3 2 1 2 1 ]
:
:         o
:  o o o ooo o
: ooooooooooooo
:
:
:   3:      [ 1 2 1 2 1 2 3 2 1 2 1 2 1 ]
:
:       o
:  o o ooo o o
: ooooooooooooo
:
:
:   4:      [ 1 2 1 2 3 2 1 2 1 2 1 2 1 ]
:
:     o
:  o ooo o o o
: ooooooooooooo
:
:
:   5:      [ 1 2 3 2 1 2 1 2 1 2 1 2 1 ]
:
:   o
:  ooo o o o o
: ooooooooooooo
:
:
:   6:      [ 1 2 3 2 3 4 3 2 1 ]
:
:      o
:   o ooo
:  ooooooo
: ooooooooo
:
:
:   7:      [ 1 2 3 4 3 2 3 2 1 ]
:
:    o
:   ooo o
:  ooooooo
: ooooooooo
(End)
From _Joerg Arndt_, Mar 09 2014: (Start)
The A097331(9) = 14 such sandpiles with base length 9 are:
01:  [ 1 2 1 2 1 2 1 2 1 ]
02:  [ 1 2 1 2 1 2 3 2 1 ]
03:  [ 1 2 1 2 3 2 3 2 1 ]
04:  [ 1 2 1 2 3 2 1 2 1 ]
05:  [ 1 2 1 2 3 4 3 2 1 ]
06:  [ 1 2 3 2 1 2 3 2 1 ]
07:  [ 1 2 3 2 1 2 1 2 1 ]
08:  [ 1 2 3 2 3 2 1 2 1 ]
09:  [ 1 2 3 2 3 2 3 2 1 ]
10:  [ 1 2 3 4 3 2 1 2 1 ]
11:  [ 1 2 3 2 3 4 3 2 1 ]
12:  [ 1 2 3 4 3 2 3 2 1 ]
13:  [ 1 2 3 4 3 4 3 2 1 ]
14:  [ 1 2 3 4 5 4 3 2 1 ]
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Joerg Arndt)
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Cf. A049346 (g.f.: 1 - 1/G(0), where G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A173258, A291874.

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 + (-q)^(k+1) / (1 - (-q)^(k+1) / G(k+1) ) );
gf = 1 / G(0);
Vec(gf)

N = 66;  q = 'q + O('q^N);
F(q,y,k) = if(k>N, 1, 1/(1 - y*q^2 * F(q, q^2*y, k+1) ) );
Vec( 1 + q * F(q,q,0) ) \\ Joerg Arndt, Mar 09 2014

a(0) = 1 and a(n) = abs(A049346(n)) for n>=1.
G.f.: 1/ (1-q/(1+q/ (1+q^2/(1-q^2/ (1-q^3/(1+q^3/ (1+q^4/(1-q^4/ (1-q^5/(1+q^5/ (1+-...) )) )) )) )) )).
G.f.: 1 + q * F(q,q) where F(q,y) = 1/(1 - y * q^2 * F(q, q^2*y) ); cf. A005169 and p. 841 of the Odlyzko/Wilf reference; 1/(1 - q * F(q,q)) is the g.f. of A143951. - Joerg Arndt, Mar 09 2014
G.f.: 1 + q/(1 - q^3/(1 - q^5/(1 - q^7/ (...)))) (from formulas above). - Joerg Arndt, Mar 09 2014
G.f.: F(x, x^2) where F(x,y) is the g.f. of A239927. - Joerg Arndt, Mar 29 2014
a(n) ~ c * d^n, where d = 1.23729141259673487395949649334678514763130846902468... and c = 0.0773368373684184197215007198148835507944051447907... - Vaclav Kotesovec, Sep 05 2017
G.f.: A(x) = 2 -1/A143951(x). - R. J. Mathar, Aug 23 2018

A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 11, 5, 2, 0, 0, 0, 0, 15, 8, 4, 0, 0, 0, 0, 0, 22, 14, 10, 3, 0, 0, 0, 0, 0, 30, 21, 18, 7, 1, 0, 0, 0, 0, 0, 42, 32, 32, 17, 6, 0, 0, 0, 0, 0, 0, 56, 45, 50, 31, 15, 2, 0, 0, 0, 0, 0, 0, 77, 65, 80, 58, 36, 11, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jul 01 2015

In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.
The diagonals T(n,n-k) count connected skew diagrams with weight k:
  1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;
Their sums equal A006958.

			T(7,2) = 4, the pairs of partitions are ((4,3)/(2)), ((3,3,1)/(2)), ((3,2,2)/(1,1)) and ((2,2,2,1)/(1,1));
The diagrams are:
  x x 0 0 , x x 0 , x 0 0 , x 0
  0 0 0     0 0 0   x 0     x 0
            0       0 0     0 0
                            0
Triangle begins:
      k=0  1  2  3  4  5  6  7
  n=0;  0
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  0  0  0
  n=4;  5  1  0  0  0
  n=5;  7  2  0  0  0  0
  n=6; 11  5  2  0  0  0  0
  n=7; 15  8  4  0  0  0  0  0

I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.

Wouter Meeussen, Table n,m, T(n,m) for n= 1..27

Cf. A259478, A259479, A259481, A161492, A227309, A006958.

(* see A259479 *) factor[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=Block[{a1,a2,a3},a1=Apply[Join,Table[{i,j},{i,Length[\[Lambda]]},{j,\[Lambda][[i]],\[Lambda][[Min[i+1,Length[\[Lambda]]]]],-1}]];
a2=Map[{First[#],First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&,a1];a3=Map[First,DeleteCases[SplitBy[a2,MatchQ[#,{,False}]&],{{,False}}],{2}];
Flatten[redu[Part[\[Lambda],#], Part[PadRight[\[Mu],Length[\[Lambda]],0],#]/. 0->Sequence[]]&/@Map[Union,a3],1]];
Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]

A259479 Skew diagrams, both connected or not.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 5, 2, 0, 0, 0, 11, 9, 6, 1, 0, 0, 0, 15, 13, 12, 6, 0, 0, 0, 0, 22, 20, 22, 14, 3, 0, 0, 0, 0, 30, 28, 36, 27, 13, 2, 0, 0, 0, 0, 42, 40, 56, 48, 31, 11, 1, 0, 0, 0, 0, 56, 54, 82, 77, 59, 33, 9, 0, 0, 0, 0, 0, 77, 75, 120, 121, 106, 72, 30, 6, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jun 28 2015

T(n,m) counts pairs of partitions lambda of n and mu of 0<=m<=n respectively, so that the Ferrers diagram of mu does not exceed that of lambda, and that the diagrams of lambda and mu do not contain equal rows or columns.

			T(6,2) = 6, the pairs of partitions are ((4,2)/(2)), ((3,3)/(2)), ((3,2,1)/(2)), ((3,2,1)/(1,1)), ((2,2,2)/(1,1)) and ((2,2,1,1)/(1,1))
and the diagrams are:
  x x 0 0 , x x 0 , x x 0 , x 0 0 , x 0 , x 0
  0 0       0 0 0   0 0     x 0     x 0   x 0
                    0       0       0 0   0
                                          0
Triangle begins:
      k=0  1  2  3  4  5  6
  n=0;  1
  n=1;  1  0
  n=2;  2  0  0
  n=3;  3  1  0  0
  n=4;  5  3  0  0  0
  n=5;  7  5  2  0  0  0
  n=6; 11  9  6  1  0  0  0

I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.

Wouter Meeussen, Table n, m, T(n,m) for n= 1..27

Cf. A259478, A259480, A259481, A161492, A227309, A006958.

majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
redu1[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=Delete[#,List/@DeleteCases[Table[i Boole[\[Lambda][[i]]==\[Mu][[i]]],{i,Length[\[Mu]]}],0]]&/@{\[Lambda],\[Mu]};
redu[\[Lambda],\[Mu]]/;majorsweak[\[Lambda],\[Mu]]:=TransposePartition/@Apply[redu1,TransposePartition/@redu1[\[Lambda],\[Mu]]];
Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}];

A161492 Triangle T(n,m) read by rows: the number of skew Ferrers diagrams with area n and m columns.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 17, 13, 5, 1, 1, 12, 32, 34, 19, 6, 1, 1, 16, 55, 78, 58, 26, 7, 1, 1, 20, 89, 160, 154, 90, 34, 8, 1, 1, 25, 136, 305, 365, 269, 131, 43, 9, 1, 1, 30, 200, 544, 794, 716, 433, 182, 53, 10, 1, 1, 36, 284, 923, 1609, 1741, 1271, 657, 244, 64, 11, 1

Offset: 1

R. J. Mathar, Jun 11 2009

Row sums give A006958, sums along falling diagonals give A227309. [Joerg Arndt, Mar 23 2014]

A coin fountain is an arrangement of coins in numbered rows such that the bottom row (row 0) contains contiguous coins and such that each coin in a higher row touches exactly two coins in the next lower row. See A005169. T(n,m) equals the number of coin fountains with exactly n coins in the even-numbered rows and n - m coins in the odd-numbered rows of the fountain. See the illustration in the Links section. - Peter Bala, Jul 21 2019

			T(4,2)=4 counts the following 4 diagrams with area equal to 4 and 2 columns:
   .X..XX...X..XX
   XX..XX...X..X.
   X.......XX..X.
From _Joerg Arndt_, Mar 23 2014: (Start)
Triangle begins:
01:  1
02:  1   1
03:  1   2    1
04:  1   4    3    1
05:  1   6    8    4     1
06:  1   9   17   13     5     1
07:  1  12   32   34    19     6     1
08:  1  16   55   78    58    26     7    1
09:  1  20   89  160   154    90    34    8   1
10:  1  25  136  305   365   269   131   43   9   1
11:  1  30  200  544   794   716   433  182  53  10  1
12:  1  36  284  923  1609  1741  1271  657 ...
(End)

Peter Bala, Illustration for the terms of row 4
Peter Bala, Fountains of coins and skew Ferrers diagrams
M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Disc. Math. (1993) Vol.112, No. 1-3, pp. 65-79.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See p. 19.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 5.

Row sums A006958. Cf. A005169, A227309.

qpoch := proc(a,q,n)
    mul( 1-a*q^k,k=0..n-1) ;
end proc:
A161492 := proc(n,m)
    local N,N2,ns ;
    N := 0 ;
    for ns from 0 to n+2 do
        N := N+ (-1)^ns *q^binomial(ns+1,2) / qpoch(q,q,ns) / qpoch(q,q,ns+1) *q^(ns+1) *t^(ns+1) ;
        N := taylor(N,q=0,n+1) ;
    end do:
    N2 := 0 ;
    for ns from 0 to n+2 do
        N2 := N2+ (-1)^ns*q^binomial(ns,2)/(qpoch(q,q,ns))^2*q^ns*t^ns ;
        N2 := taylor(N2,q=0,n+1) ;
    end do:
    coeftayl(N/N2,q=0,n) ;
    coeftayl(%,t=0,m) ;
end proc:
for a from 1 to 20 do
    for c from 1 to a do
        printf("%d ", A161492(a,c)) ;
    od:
od:

nmax = 13;
qn[n_] := Product[1 - q^k, {k, 1, n}];
nm = Sum[(-1)^n q^(n(n+1)/2)/(qn[n] qn[n+1])(t q)^(n+1) + O[q]^nmax, {n, 0, nmax}];
dn = Sum[(-1)^n q^(n(n-1)/2)/(qn[n]^2)(t q)^n + O[q]^nmax, {n, 0, nmax}];
Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[nm/dn, q]] // Flatten (* Jean-François Alcover, Dec 19 2019, after Joerg Arndt *)

/* formula from the Delest/Fedou reference: */
N=20;  q='q+O('q^N);  t='t;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
v=Vec(nm/dn);
for(n=1,N-1,print(Vec(polrecip(Pol(v[n])))));  \\ print triangle
\\ Joerg Arndt, Mar 23 2014

From Peter Bala, Jul 21 2019: (Start)
The following formulas all include an initial term T(0,0) = 1.
O.g.f. as a ratio of q-series: A(q,t) = N(q,t)/D(q,t) = 1 + q*t + q^2*(t + t^2) + q^3*(t + 2*t^2 + t^3) + ..., where N(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2 and D(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2.
Continued fraction representations:
A(q,t) = 1/(1 - q*t/(1 - q/(1 - q^2*t/(1 - q^2/(1 - q^3*t/(1 - q^3/(1 - (...) ))))))).
A(q,t) = 1/(1 - q*t/(1 + q*(t - 1) - q*t/(1 + q*(t - q) - q*t/( 1 + q*(t - q^2) - q*t/( (...) ))))).
A(q,t) = 1/(1 - q*t - q^2*t/(1 - q*(1 + q*t) - q^4*t/(1 - q^2*(1 + q*t) - q^6*t/(1 - q^3*(1 + q*t) - q^8*t/( (...) ))))).
A(q,t) =  1/(1 - q*t - q^2*t/(1 - q^2*t - q/(1 - q^3*t - q^5*t/(1 - q^4*t - q^2/(1 - q^5*t - q^8*t/ (1 - q^6*t - q^3/(1 - q^7*t - q^11*t/(1 - q^8*t - (...) )))))))). (End)

A067675 Number of fixed convex polyominoes with n cells.

Original entry on oeis.org

1, 2, 6, 19, 59, 176, 502, 1374, 3630, 9312, 23320, 57279, 138536, 331032, 783630, 1841867, 4306172, 10028276, 23288394, 53974959, 124925967, 288878550, 667602492, 1542254655, 3562000916, 8225719574, 18994263354, 43858728367, 101270779744, 233836327750, 539935689810

Offset: 1

Steven Finch, Feb 04 2002

nonn

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.19, p. 380.

Ruben Grønning Spaans, Table of n, a(n) for n = 1..1000
Gadi Aleksandrowicz, Andrei Asinowski and Gill Barequet, A polyominoes-permutations injection and tree-like convex polyominoes, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, Pages 503-520.
M. Bousquet-Mélou and J.-M. Fédou, The generating function of convex polyominoes: the resolution of a q-differential system, Discrete Math. 137 (1995) 53-75.
Ruben Grønning Spaans, C program.
V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian).

Cf. A006958, A067676 (fixed directed convex polyominoes), A191148 (fixed line-convex polycubes in 3 dimensions), A276994.

a(n) ~ c * d^n, where d = A276994 = 2.309138593330494731098720305017212531911814472581628401694402900284456440748..., c = 2.91959850971360705538470951565133568591516894147305658630679268977185945... . - Vaclav Kotesovec, Sep 27 2016

Six more terms from Ruben Grønning Spaans, Sep 20 2014

A300120 Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

Original entry on oeis.org

2, 6, 12, 26, 44, 86, 136, 239, 376, 613, 930, 1485, 2194, 3355, 4948, 7372, 10656, 15660, 22359, 32308

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(3) = 12 skew partitions:
(3)/()   (3)/(1)   (3)/(2)    (3)/(3)
(21)/()  (21)/(11) (21)/(2)   (21)/(21)
(111)/() (111)/(1) (111)/(11) (111)/(111)

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A297388, A299925, A299926, A300118, 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]]

cp's OEIS Frontend

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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A259478 Partition containment triangle.

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

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A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

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A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

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A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).

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A259479 Skew diagrams, both connected or not.

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