A105124 -id:A105124 - cp's OEIS frontend

A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 45, 197, 45, 1, 1, 1, 1, 197, 4593, 4593, 197, 1, 1, 1, 1, 903, 126289, 732963, 126289, 903, 1, 1, 1, 1, 4279, 3888343, 155242003, 155242003, 3888343, 4279, 1, 1, 1, 1, 20793, 130016393, 40007492715, 289599115433, 40007492715, 130016393, 20793, 1, 1

Offset: 0

Andrew Howroyd, Sep 16 2024

T(n,k) is the number of normal generalized Young tableaux with all rows and columns strictly increasing whose shape is a rectangle of size n X k (cf. A299968). - Ludovic Schwob, Nov 18 2024

			Array begins:
=====================================================================
n/k | 0 1   2       3           4               5               6 ...
----+----------------------------------------------------------------
  0 | 1 1   1       1           1               1               1 ...
  1 | 1 1   1       1           1               1               1 ...
  2 | 1 1   3      11          45             197             903 ...
  3 | 1 1  11     197        4593          126289         3888343 ...
  4 | 1 1  45    4593      732963       155242003     40007492715 ...
  5 | 1 1 197  126289   155242003    289599115433 723253222084867 ...
  6 | 1 1 903 3888343 40007492715 723253222084867 ...
...
The T(2,3) = 11 matrices are:
  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 4]  [1 2 4]
  [2 3 4]  [2 4 5]  [3 4 5]  [4 5 6]  [2 3 5]  [3 4 5]
.
  [1 2 4]  [1 2 5]  [1 3 4]  [1 3 4]  [1 3 5]
  [3 5 6]  [3 4 6]  [2 4 5]  [2 5 6]  [2 4 6]

Andrew Howroyd, Table of n, a(n) for n = 0..230
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54.

Columns k=1..4 are A000012, A001003, A105124, A374985.
Main diagonal is A374514.
Cf. A060854 (case all values also distinct), A299968.

\\ See PARI link in A374514 for program code.
for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

T(n,k) = T(k,n).

A088594 Number of 3-dimensional lattice paths running from (0,0,0) to (n,n,n), lying in {(x,y,z) : 0<=x<=y<=z} and using the steps (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1).

Original entry on oeis.org

1, 4, 44, 788, 18372, 505156, 15553372, 520065572, 18518471492, 692900847812, 26985709712524, 1086313382608436, 44960426477218436, 1905328431907938180, 82405332511166288572, 3627806131038258219076, 162218975410046793174404

Offset: 0

Robert A. Sulanke (sulanke(AT)math.boisestate.edu), Nov 20 2003

nonn

These are the 3-dimensional analogs of the large Schröder numbers, A006318.
R(3,n) = 4*A105124(n) for n>0, where A105124 is the three-dimensional small Schröder numbers. - Paul D. Hanna, Apr 19 2005
Number of n X 3 semi-standard Young tableaux with consecutive entries. I.e., if j is in P, and 1<=i<=j, then i is in P. - Graham H. Hawkes, Feb 16 2015

Alois P. Heinz, Table of n, a(n) for n = 0..200
R. A. Sulanke, Three-dimensional Narayana and Schröder numbers
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16).

Cf. A006318, A001003, A087647.

1, seq( add( add(2*(-1)^(k-j)*binomial(3*n+1, k-j)* binomial(n+j,n)*binomial(n+j+1,n)*binomial(n+j+2,n)/(n+1)^2/(n+2), j = 0 .. k) *2^(k+2), k = 0 .. 2*n-2), n = 1 ..20 );

Flatten[{1,Table[Sum[Sum[2*(-1)^(k-j)*Binomial[3*n+1,k-j]*Binomial[n+j,n]*Binomial[n+j+1,n]*Binomial[n+j+2,n]/(n+1)^2/(n+2),{j,0,k}]*2^(k+2),{k,0,2*n-2}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 14 2013 *)

{alias(C,binomial); R3(n)=if(n==0,1,sum(k=0,2*n-2, 2^(k+2)*sum(j=0,k, 2*(-1)^(k-j)*C(3*n+1,k-j)*C(n+j,n)*C(n+j+1,n)*C(n+j+2,n)/(n+1)^2/(n+2))))} \\ Paul D. Hanna, Apr 19 2005

For n => 1, R(3, n) := Sum[2^(k+2)*Sum[2*(-1)^(k-j)*C(3*n+1, k-j)* C(n+j, n)*C(n+j+1, n)*C(n+j+2, n)/(n+1)^2/(n+2), {j, 0, k}], {k, 0, 2*n-2}]. For n => 4, (3n-4)(n+2)(n+1)^2 R(3, n)(t) = (3n-2)(n+1)( 4(1+t+t^2) - 5(1+7t+t^2)n+3(1+7t+t^2)n^2 )R(3, n-1) - (n-2)( -12 +29n -30n^2 +9n^3)(1-t)^4 R(3, n-2) + (3n-1)(n-2)(n-3)(n-4) (1-t)^6 R(3, n-3).
G.f.: (1+2/x)*(1-1/x)*Int(((x-1)*(7*x^3-12*x^2+57*x+2)*hypergeom([1/3, 2/3],[1],54*x/(1-x)^3)-x*(x+5)*(x^2-8*x-11)*hypergeom([2/3, 4/3],[2],54*x/(1-x)^3))/(3*(x-1)^4*(x+2)^2),x)-(1+4*x)/(3*x). - Mark van Hoeij, Apr 16 2013
Recurrence: (n+1)^2*(n+2)*(3*n-4)*a(n) = (n+1)*(3*n-2)*(57*n^2 - 95*n + 28)*a(n-1) - (n-2)*(9*n^3 - 30*n^2 + 29*n - 12)*a(n-2) + (n-4)*(n-3)*(n-2)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Aug 14 2013
a(n) ~ c*d^n/n^4, where d = 12*2^(2/3)+15*2^(1/3)+19 = 56.947628372... is the root of the equation d^3-57*d^2+3*d-1=0 and c = sqrt(4 + 10*2^(1/3)/3 + 8*2^(2/3)/3)/Pi = 1.122366540310337391196984583368763794289876... - Vaclav Kotesovec, Aug 14 2013, updated Mar 19 2015
In general, the number of SSYT of shape n X d with consecutive entries is given by:
[Prod_(i=1,d-1) (i/(n+i))^(d-1)] *
Sum_(j=0,n*(d-1)) [Prod_(i=0,d-1) (n+i+j choose n)*(Sum_(k=0,(n*(d-1)-j)) (-1)^k (n+j+k choose k)]. - Graham H. Hawkes, Feb 16 2015

A375052 Number of n X 4 matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

Original entry on oeis.org

1, 1, 45, 4593, 732963, 155242003, 40007492715, 11910942902211, 3961395353371797, 1438326441446892453, 560815045483180502313, 231969721919492199940197, 100843924627856371008805767, 45742671500990337278105740455, 21524779658986553968372985081175

Offset: 0

Andrew Howroyd, Sep 16 2024

nonn

In other words, a(n) is the number of increasing tableaux of shape (n,n,n,n).

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=4 of A374985.

Cf. A105124.

cp's OEIS Frontend

A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A088594 Number of 3-dimensional lattice paths running from (0,0,0) to (n,n,n), lying in {(x,y,z) : 0<=x<=y<=z} and using the steps (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A375052 Number of n X 4 matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

Views

Author

Keywords

Comments

Links

Crossrefs