A184173 -id:A184173 - cp's OEIS frontend

A005157 Number of totally symmetric plane partitions that fit in an n X n X n box.

1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, 1885102080, 62062015500, 2652584509440, 147198472495020, 10606175914819584, 992340657705109416, 120567366227960791040, 19023173201224270401428, 3897937005297330777227264

Offset: 0

N. J. A. Sloane

Also, number of 2-dimensional shifted complexes on n+1 nodes. [Klivans]
Also the number of totally symmetric partitions which fit in an (n-1)-dimensional box with side length 4 (for n>0). - Graham H. Hawkes, Jan 11 2014
Suppose we index this sequence slightly differently. Let the elements of a partition be represented by points rather than boxes, as in a Ferrers diagram. In this case, a 1 X 1 X 1 (closed) box would fit 8 points -- one at each vertex of the box, and we use the convention that a 0 X 0 X 0 (closed) box contains exactly one point. Using this indexing, the sequence begins (offset is still 0) 2,5,16,... rather than 1,2,5,... If we use the same method of indexing for all other dimensions, then we have the following remarkable result: The number of totally symmetric partitions which fit inside a d-dimensional box with side length n is equal to the number of totally symmetric partitions which fit inside an n-dimensional box of side length d. - Graham H. Hawkes, Jan 11 2014
For two other contexts where this sequence arises, see the Knuth (2019) link (noncrossing paths among the 2(2^n-1) paths defined in that note; independent sets of paths among the first 2^n-1 of those). - N. J. A. Sloane, Feb 09 2019, based on email from Don Knuth.

			a(2) = 5 because we have: void, 1, 21/1, 22/21, and 22/22.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198 (corrected).
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..130 (first 51 terms from T. D. Noe)
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
R. K. Guy, Letter to N. J. A. Sloane, Dec 5 1988.
Graham H. Hawkes, Totally symmetric partitions in boxes
Seth Ireland, A bijection between strongly stable and totally symmetric partitions, arXiv:2302.02505 [math.CO], 2023.
C. Klivans, Obstructions to shiftedness, preprint.
C. Klivans, Obstructions to shiftedness, Discrete Comput. Geom., 33 (2005), 535-545.
Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Cf. A049505, A184173, A323848.

Row sums of A214564.

A005157 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+2*j-2),j=i..n),i=1..n); end;

Table[Product[(i+j+k-1)/(i+j+k-2),{i,n},{j,i,n},{k,j,n}],{n,0,20}] (* Harvey P. Dale, Jul 17 2011 *)

A005157(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+2*j-2))) \\ M. F. Hasler, Sep 26 2018

a(n) = Product_{i=1..n} Product_{j=i..n} Product_{k=j..n} (i+j+k-1)/(i+j+k-2). - Paul Barry, May 13 2008
a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n*(n+1)/4 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(n*(2*n+1)/2 + 13/24)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015
a(n) = sqrt(A323848(n+1,n)) for n >= 1. [proof by Nikolai Beluhov; see Knuth (2019) link] - Alois P. Heinz, Feb 10 2019
Apparently, a(n) = Sum_{k=0..n} A184173(n,k). - Alois P. Heinz, Feb 11 2019
Conjectures: if p == 1 (mod 6) is prime then a(p) == 2^((p+5)/6) (mod p^2); if p == 5 (mod 6) is prime then a(p) == 2^((p+1)/6) (mod p^2) (checked up to p = 1009). - Peter Bala, Feb 17 2023

A306376 Sum of the 2 X 2 minors in the n X n Pascal matrix.

Original entry on oeis.org

0, 0, 1, 7, 34, 144, 574, 2226, 8533, 32587, 124453, 476145, 1826175, 7022379, 27072487, 104614863, 405122290, 1571859864, 6109296442, 23781666198, 92704406320, 361832294964, 1413879679672, 5530590849168, 21654384302110, 84859670743770, 332818903663390

Offset: 0

Alois P. Heinz, Feb 11 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1664
Wikipedia, Minor (linear algebra)

Column k=2 of A184173.

Cf. A007318, A369906.

a:= proc(n) option remember; `if`(n<3, (n-1)*n/2,
     ((7*n^2-16*n+6)*a(n-1)-2*(7*n^2-17*n+9)*a(n-2)+
      4*(n-1)*(2*n-3)*a(n-3))/(n*(n-2)))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n < 3, (n-1)n/2,
     ((7n^2 - 16n + 6) a[n-1] - 2(7n^2 - 17n + 9) a[n-2] +
     4(n-1)(2n-3) a[n-3])/(n(n-2))];
a /@ Range[0, 30] (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)

G.f.: -1/(2*(x-1))*(1/(2*x-1)+1/sqrt(1-4*x)).

a(n) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 19 2024

A369559 T(n,k) is the sum of the permanents of all k X k submatrices in the n X n Pascal matrix; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 9, 1, 1, 15, 50, 35, 1, 1, 31, 234, 482, 185, 1, 1, 63, 1016, 5011, 6894, 1267, 1, 1, 127, 4256, 46252, 162724, 150624, 10633, 1, 1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1, 1, 511, 71349, 3415771, 59157822, 362855438, 696003275, 223813933, 1196889, 1

Offset: 0

Alois P. Heinz, Jan 25 2024

			T(3,2) = 9:
  The 3 X 3 Pascal matrix
    [1 0 0]
    [1 1 0]
    [1 2 1]
  has nine 2 X 2 submatrices
    [1 0] [1 0] [0 0] [1 0] [1 0] [0 0] [1 1] [1 0] [1 0]
    [1 1] [1 0] [1 0] [1 2] [1 1] [2 1] [1 2] [1 1] [2 1].
  Sum of their permanents is 1 + 0 + 0 + 2 + 1 + 0 + 3 + 1 + 1 = 9.
Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,     1;
  1,   7,     9,      1;
  1,  15,    50,     35,       1;
  1,  31,   234,    482,     185,       1;
  1,  63,  1016,   5011,    6894,    1267,       1;
  1, 127,  4256,  46252,  162724,  150624,   10633,      1;
  1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1;
  ...

Wikipedia, Permanent (mathematics)

Columns k=0-2 give: A000012, A000225, A369906.

Cf. A007318, A184173 (same for determinants).

with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(k=0 or k=n, 1, (l-> add(add(
      Permanent(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
       i, j)), j in l), i in l))(choose([$1..n], k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..9);

T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
    l = Subsets[Range[n], {k}];
    M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
    Total[Permanent /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 29 2024 *)

A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 13, 1, 1, 15, 65, 73, 21, 1, 1, 21, 140, 273, 171, 31, 1, 1, 28, 266, 798, 871, 346, 43, 1, 1, 36, 462, 1974, 3321, 2306, 631, 57, 1, 1, 45, 750, 4326, 10377, 11126, 5335, 1065, 73, 1, 1, 55, 1155, 8646, 28017, 42878, 31795, 11145, 1693, 91, 1

Offset: 0

Vladimir Kruchinin, Mar 19 2025

Triangle T(n,k) of minors of the main diagonal of Pascal's matrix, n -size matrix, k - number of element of diagonal.

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,   1;
  1, 10,  25,  13,   1;
  1, 15,  65,  73,  21,  1;
  1, 21, 140, 273, 171, 31, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Wikipedia, Minor (linear algebra)

Columns k=0-3 give: A000012, A000217, A001296(n-1) for n>=1, A107963(n-3) for n>=3.
Row sums give A024718.
T(n+1,n) gives A002061(n+1).
Cf. A007318, A184173.

T:= proc(n, k) option remember; `if`(n<0, 0,
      T(n-1, k)+binomial(n-1, k-1)*binomial(n, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 20 2025

A382225[n_, k_] := A382225[n, k] = If[k == n, 1, A382225[n-1, k] + Binomial[n-1, k-1]*Binomial[n, k]];
Table[A382225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 22 2025 *)

h[i,j]:=binomial(i+j-3,i-1);
for n:1 thru 7 do
    if n=1 then print([1])
    else (M:genmatrix(h,n,n),
          print(makelist(determinant(minor(M,k,k)),k,1,n))
         );

G.f.: 1/(1-x) * ((1-x*(1-y))/(2*(sqrt((1-x*(1+y))^2-4*x^2*y)))+1/2).

T(n,k) = T(n-1,k)+C(n-1,k-1)*C(n,k).

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A005157 Number of totally symmetric plane partitions that fit in an n X n X n box.

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A306376 Sum of the 2 X 2 minors in the n X n Pascal matrix.

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A369559 T(n,k) is the sum of the permanents of all k X k submatrices in the n X n Pascal matrix; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

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