A161492 -id:A161492 - cp's OEIS frontend

A006958 Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused).

1, 2, 4, 9, 20, 46, 105, 242, 557, 1285, 2964, 6842, 15793, 36463, 84187, 194388, 448847, 1036426, 2393208, 5526198, 12760671, 29466050, 68041019, 157115917, 362802072, 837759792, 1934502740, 4467033943, 10314998977, 23818760154, 55000815222, 127004500762

Offset: 1

N. J. A. Sloane, Simon Plouffe

Same as: number of skew Ferrers diagrams. - Joerg Arndt, Mar 18 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. a(n) equals the number of coin fountains with exactly n coins in the even-numbered rows of the fountain. See the illustration in the Links section. See A161492 for a refinement of this sequence. - Peter Bala, Jul 20 2019

			G.f. may be expressed by the continued fraction: 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-x^4/(1-...)))))))) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 105*x^7 + ...
From _Michael B. Porter_, Sep 21 2016; corrected by _Riccardo Moschetti_, Aug 11 2017: (Start)
Here are the nine parallelogram polyominoes with 4 cells, i.e., polygons convex according to the -45-degree direction, according to "Polya Festoons" of P. Flajolet:
                          _      _  _
             _  _     _ /_ /   /_ /_ /
         _ /_ /_ /  /_ /_ /   /_ /      _  _  _  _
       /_ /_ /     /_ /      /_ /     /_ /_ /_ /_ /
                     _
              _    /_ /    _  _  _            _  _
            /_ /  /_ /   /_ /_ /_ /    _    /_ /_ /
         _ /_ /  /_ /   /_ /    _  _ /_ /  /_ /_ /
       /_ /_ /  /_ /          /_ /_ /_ /
(End)

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.19, p. 380.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..2752 (terms 1..1000 from Robert Israel)
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Peter Bala, Illustration for the initial terms of the sequence.
Peter Bala, Fountains of coins and skew Ferrers diagrams.
E. A. Bender, Convex n-ominoes, Discrete Math., 8 (1974), 219-226.
M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics, vol. 112 (1993), no. 1-3, pp. 65-79.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
P. Flajolet, Email to N. J. A. Sloane & S. Plouffe, Aug. 1991.
P. Flajolet, Polya Festoons, INRIA Research Report, No 1507, September 1991. 6pp.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 661.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 4-5.
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. 2.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math., 8 (1974), 31-40.

Cf. A075125, A276994, A275761, A275762, A161492.

N:= 100: # to get a(1) to a(N)
M:= ceil(sqrt(N+1)):
C:= 1:
for j from M to 1 by -1 do C:= 1/(1-x^j/(1-x^j*C)) od:
S:= series(C,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Sep 20 2016

NN = 100; (* to get a(1) to a(NN) *) M = Ceiling[Sqrt[NN+1]]; c = 1; For[j = M, j >= 1, j--, c = 1/(1-x^j/(1-x^j*c))]; c = Series[c, {x, 0, NN+1}]; CoefficientList[c, x][[2 ;; NN+1]] (* Jean-François Alcover, Sep 27 2016, adapted from Robert Israel's Maple code *)
nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1] - Floor[Range[nmax + 1]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

{a(n)=local(CF=1+x*O(x^n),m); for(k=0,n\2,m=n\2-k+1;CF=(1-x^((m+1)\2)/CF));polcoeff(1/CF,n)} \\ Paul D. Hanna, May 14 2005

/* From the Delest/Fedou reference: */
N=44;  q='q+O('q^N);  t=1;
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 );
Vec(nm/dn)  \\ Joerg Arndt, Mar 18 2014

G.f.: 1+A(x) = 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-...))))))) (continued fraction). - Paul D. Hanna, May 14 2005
The continued fraction given by P. Flajolet, "Polya Festoons", is derived from a q-expansion, C(x, y;q), where C(1, 1;q) = q/(1-2*q-q^3/(1-2*q^2-q^5/(1-2*q^3-q^7/(1-2*q^4-q^9/(1-...))))) = q + 2*q^2 + 4*q^3 + 9*q^4 + 20*q^5 + 46*q^6 + 105*q^7 + ... - Paul D. Hanna, May 14 2005
G.f.: 1/x/G(0) -1/x, where G(k)= 1 - x^(k+1)/(1 - x^(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x^(k+1)/( x^(k+1) - 1/(1 - x^(k+1)/( x^(k+1) - 1/W(k+1) ))); R=1 (continued fraction). - Sergei N. Gladkovskii, Aug 27 2013
a(n) ~ c * d^n, where d = A276994 = 2.309138593330494731098720305017212531911814472581628401694402900284456440748..., c = 0.29745350581112195107675842441785013227507248969090226252518932405713367... . - Vaclav Kotesovec, Sep 21 2016
From Peter Bala, Jul 21 2019: (Start)
O.g.f. as a ratio of q-series: 1 + A(q) = N(q)/D(q) = 1 + q + 2*q^2 + 4*q^3 + ..., where N(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)/Product_{k = 1..n} (1 - q^k)^2 and D(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)/Product_{k = 1..n} (1 - q^k)^2.
The constant d = 2.30913... in the above asymptotic formula is a zero of D(q) (as is 1/d).
Continued fraction representations for the o.g.f.:
1 + A(q) = 1/(1 - q/(1 - q/(1 + q*(1 - q) - q/(1 + q*(1 - q^2) - q/(1 + q*(1 - q^3) - (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q*(1 + q) - q^4/(1 - q^2*(1 + q) - q^6/(1 - q^3(1 + q) - q^8/( (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q^2 - q/(1 - q^3 - q^5/(1 - q^4 - q^2/(1 - q^5 - q^8/(1 - q^6 - q^3/(1 - q^7 - q^11/(1 - q^8 - (...) )))))))). (End)

More terms from Paul D. Hanna, May 14 2005

Definition modified by Don Knuth, Sep 20 2016

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

A047998 Triangle of numbers a(n,k) = number of "fountains" with n coins, k in the bottom row.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 3, 6, 5, 1, 0, 0, 0, 0, 2, 7, 10, 6, 1, 0, 0, 0, 0, 1, 7, 14, 15, 7, 1, 0, 0, 0, 0, 1, 5, 17, 25, 21, 8, 1, 0, 0, 0, 0, 0, 5, 16, 35, 41, 28, 9, 1, 0, 0, 0, 0, 0, 3, 16, 40, 65, 63, 36, 10, 1, 0, 0, 0, 0, 0, 2, 14, 43, 86, 112, 92, 45, 11, 1, 0, 0, 0, 0, 0, 1, 11, 44, 102, 167, 182, 129, 55, 12, 1, 0, 0, 0, 0, 0, 1, 9, 40, 115, 219, 301, 282, 175, 66, 13, 1

Offset: 0

N. J. A. Sloane

The number a(n,k) of (n,k) fountains equals the number of 231-avoiding permutations in the symmetric group S_{k} with exactly n - k inversions (Brändén et al., Proposition 4).

			Triangle begins:
00:  1;
01:  0,1;
02:  0,0,1;
03:  0,0,1,1;
04:  0,0,0,2,1;
05:  0,0,0,1,3,1;
06:  0,0,0,1,3,4,1;
07:  0,0,0,0,3,6,5,1;
08:  0,0,0,0,2,7,10,6,1;
09:  0,0,0,0,1,7,14,15,7,1;
10:  0,0,0,0,1,5,17,25,21,8,1;
11:  0,0,0,0,0,5,16,35,41,28,9,1;
12:  0,0,0,0,0,3,16,40,65,63,36,10,1;
13:  0,0,0,0,0,2,14,43,86,112,92,45,11,1;
14:  0,0,0,0,0,1,11,44,102,167,182,129,55,12,1;
15:  0,0,0,0,0,1,9,40,115,219,301,282,175,66,13,1;
16:  0,0,0,0,0,0,7,37,118,268,434,512,420,231,78,14,1;
17:  0,0,0,0,0,0,5,32,118,303,574,806,831,605,298,91,15,1;
...
From _Joerg Arndt_, Mar 25 2014: (Start)
The compositions (compositions starting with part 1 and up-steps <= 1) corresponding to row n=8 with their base lengths are:
01:    [ 1 2 3 2 ]               4
02:    [ 1 2 2 3 ]               4
03:    [ 1 2 3 1 1 ]             5
04:    [ 1 2 2 2 1 ]             5
05:    [ 1 1 2 3 1 ]             5
06:    [ 1 2 2 1 2 ]             5
07:    [ 1 2 1 2 2 ]             5
08:    [ 1 1 2 2 2 ]             5
09:    [ 1 1 1 2 3 ]             5
10:    [ 1 2 2 1 1 1 ]           6
11:    [ 1 2 1 2 1 1 ]           6
12:    [ 1 1 2 2 1 1 ]           6
13:    [ 1 2 1 1 2 1 ]           6
14:    [ 1 1 2 1 2 1 ]           6
15:    [ 1 1 1 2 2 1 ]           6
16:    [ 1 2 1 1 1 2 ]           6
17:    [ 1 1 2 1 1 2 ]           6
18:    [ 1 1 1 2 1 2 ]           6
19:    [ 1 1 1 1 2 2 ]           6
20:    [ 1 2 1 1 1 1 1 ]         7
21:    [ 1 1 2 1 1 1 1 ]         7
22:    [ 1 1 1 2 1 1 1 ]         7
23:    [ 1 1 1 1 2 1 1 ]         7
24:    [ 1 1 1 1 1 2 1 ]         7
25:    [ 1 1 1 1 1 1 2 ]         7
26:    [ 1 1 1 1 1 1 1 1 ]       8
There are none with base length <= 3, two with base length 4, etc., giving row 8 [0,0,0,0,2,7,10,6,1].
(End)

B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.
R. K. Guy, personal communication to N. J. A. Sloane.
See A005169 for further references.

Alois P. Heinz, Rows n = 0..200, flattened
P. Brändén, A. Claesson, E. Steingrímsson, Catalan continued fractions and increasing subsequences in permutations, Discrete Mathematics, Vol. 258, Issues 1-3, Dec. 2002, 275-287.
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Row sums give A005169 (set x=1 in the g.f.).
Column sums give A000108 (set y=1 in the g.f.). - Joerg Arndt, Mar 25 2014
T(2n+1,n+1) gives A058300(n). - Alois P. Heinz, Jun 24 2015
Cf. A161492.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add(b(n-j, j)*x, j=1..min(i+1, n))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Oct 05 2017

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j]*x, {j, 1, Min[i+1, n]}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jul 11 2018, after Alois P. Heinz *)

N=22; x='x+O('x^N);
G(k)=if (k>N, 1, 1/(1-y*x^k*G(k+1)));
V=Vec( G(1) );
my( N=#V );
rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
for(n=1, N, print( rvec( V[n]) ) ); \\ print triangle
\\ Joerg Arndt, Mar 25 2014

G.f.: 1/(1 - y*x / (1 - y*x^2 / (1 - y*x^3 / ( ... )))), from the Odlyzko/Wilf reference. - Joerg Arndt, Mar 25 2014

G.f.: ( Sum_{n >= 0} (-y)^n*x^(n*(n+1))/Product_{k = 1..n} (1 - x^k) )/ ( Sum_{n >= 0} (-y)^n*x^(n^2)/Product_{k = 1..n} (1 - x^k) ) = 1 + y*x + y^2*x^2 + (y^2 + y^3)*x^3 + (2*y^3 + y^4)*x^4 + ... (see Berndt, Cor. to Entry 15, ch. 16). - Peter Bala, Jun 20 2019

More terms from Joerg Arndt, Mar 08 2011

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

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}];

A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 6, 3, 2, 0, 0, 0, 0, 7, 4, 4, 0, 0, 0, 0, 0, 8, 5, 6, 3, 0, 0, 0, 0, 0, 9, 6, 8, 6, 1, 0, 0, 0, 0, 0, 10, 7, 10, 9, 6, 0, 0, 0, 0, 0, 0, 11, 8, 12, 12, 11, 2, 0, 0, 0, 0, 0, 0, 12, 9, 14, 15, 16, 9, 2, 0, 0, 0, 0, 0, 0, 13, 10, 16, 18, 21, 16, 7, 0, 0, 0, 0, 0, 0, 0, 14, 11, 18, 21, 26, 23, 18, 4, 0, 0, 0, 0, 0, 0, 0, 15, 12, 20, 24, 31, 30, 29, 12, 3, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Wouter Meeussen, Jul 01 2015

Border strips are defined as connected skew tabloids free of 2-by-2 cells.

Row sums are the partition numbers (A000041), diagonals sum to 2^n (A000079).

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

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

Cf. A259478, A259479, A259480, A161492, A227309, A006958.

(* see A259479 *) Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&( Tr[\[Lambda]]-Tr[\[Mu]]==Length[\[Lambda]]+First[\[Lambda]]-1 )&& redu[\[Lambda],\[Mu]]==factor[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}]

cp's OEIS Frontend

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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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A259481 T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).