A316674 -id:A316674 - cp's OEIS frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 1, 1, 1, 75, 409, 63, 1, 1, 1, 541, 23917, 16081, 321, 1, 1, 1, 4683, 2244361, 10681263, 699121, 1683, 1, 1, 1, 47293, 308682013, 14638956721, 5552351121, 32193253, 8989, 1, 1, 1, 545835, 58514835289, 35941784497263, 117029959485121, 3147728203035, 1538743249, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2015

Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).
Row r > 0 is asymptotic to sqrt(r*Pi) * (r^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)), or equivalently to sqrt(r) * (r^(r-1)/(r-1)!)^n * (n!)^r / (2^r * (Pi*n)^((r-1)/2) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016
From Vaclav Kotesovec, Mar 23 2016: (Start)
Column k > 0 is asymptotic to sqrt(c(k)) * d(k)^n / (Pi*n)^((k-1)/2), where c(k) and d(k) are roots of polynomial equations of degree k, independent on n.
---------------------------------------------------
k               d(k)
---------------------------------------------------
2              5.8284271247461900976033774484193...
3             56.9476283720414911685286267804411...
4            780.2794068067951456595241495989622...
5          13755.2719024115081712083954421541320...
6         296476.9162644200814909862281498491264...
7        7553550.6198338218721069097516499501996...
8      222082591.6017202421029000117685530884167...
9     7400694480.0494436216324852038000444393262...
10  275651917450.6709238286995776605620357737005...
---------------------------------------------------
d(k) is a root of polynomial:
---------------------------------------------------
k=2, 1 - 6*d + d^2
k=3, -1 + 3*d - 57*d^2 + d^3
k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4
k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5
k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6
k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7
k=8, 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8
(End)
d(k) = (2^(1/k) - 1)^(-k). - David Bevan, Apr 07 2022
d(k) is asymptotic to (k/log(2))^k/sqrt(2). - David Bevan, Apr 07 2022
A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column. - Andrew Howroyd, Jan 23 2020

			A(2,2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].
Square array A(n,k) begins:
  1, 1,    1,        1,             1,                   1, ...
  1, 1,    3,       13,            75,                 541, ...
  1, 1,   13,      409,         23917,             2244361, ...
  1, 1,   63,    16081,      10681263,         14638956721, ...
  1, 1,  321,   699121,    5552351121,     117029959485121, ...
  1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...

Alois P. Heinz, Antidiagonals n = 0..48, flattened
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Columns: A000012 (k=0 and k=1), A001850 (k=2), A126086 (k=3), A263064 (k=4), A263065 (k=5), A263066 (k=6), A263067 (k=7), A263068 (k=8), A263069 (k=9), A263070 (k=10).
Rows: A000012 (n=0), A000670 (n=1), A055203 (n=2), A062208 (n=3), A062205 (n=4), A263061 (n=5), A263062 (n=6), A062204 (n=7), A263063 (n=8), A263071 (n=9), A263072 (n=10).
Main diagonal: A262810.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.
Cf. A188392, A330942, A331461, A331637.

A:= (n, k)-> add(add((-1)^i*binomial(j, i)*
     binomial(j-i, n)^k, i=0..j), j=0..k*n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[, 0] =  1; A[n, k_] := Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 22 2016, after Alois P. Heinz *)

T(n,k) = {my(m=n*k); sum(j=0, m, binomial(j,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020

A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.
A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - Peter Bala, Jan 30 2018
A(n,k) = Sum_{j=0..n*k} binomial(j,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020

A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 14, 4, 1, 1, 8, 150, 128, 8, 1, 1, 16, 2210, 10848, 1288, 16, 1, 1, 32, 41642, 1796408, 933448, 13472, 32, 1, 1, 64, 956878, 491544512, 1852183128, 85862144, 143840, 64, 1, 1, 128, 25955630, 200901557728, 7805700498776, 2098614254048, 8206774496, 1556480, 128, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

A(n,k) is the number of n-uniform k-block multisets of multisets.

			Array begins:
====================================================================
n\k | 0  1      2          3                4                  5
----+---------------------------------------------------------------
  0 | 1  1      1          1                1                  1 ...
  1 | 1  1      2          4                8                 16 ...
  2 | 1  2     14        150             2210              41642 ...
  3 | 1  4    128      10848          1796408          491544512 ...
  4 | 1  8   1288     933448       1852183128      7805700498776 ...
  5 | 1 16  13472   85862144    2098614254048 140102945876710912 ...
  6 | 1 32 143840 8206774496 2516804131997152 ...
     ...
The A(2,2) = 14 matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]  [1 0]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]  [0 2]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [2 1]  [2 0]  [1 1]  [1 0]  [2 2]
  [0 2]  [0 1]  [0 1]  [0 2]  [1 1]  [1 2]
  [1 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..2 are A000012, A121227.
Columns k=0..2 are A000012, A011782, A331397.
The version with binary entries is A330942.
The version with distinct columns is A331278.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A316674 (all), A331568 (distinct rows).
Up to row permutation: A219727, A219585, A331161, A331160.
Up to column permutation: this sequence, A331572, A331278, A331570.
Nonisomorphic: A331485.
Cf. A317583.

T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331278(n, j).
A(n, k) = A011782(n) * A330942(n, k) for k > 0.
A317583(n) = Sum_{d|n} A(n/d, d).

A052141 Number of paths from (0,0) to (n,n) that always move closer to (n,n) (and do not pass (n,n) and backtrack).

Original entry on oeis.org

1, 3, 26, 252, 2568, 26928, 287648, 3112896, 34013312, 374416128, 4145895936, 46127840256, 515268544512, 5775088103424, 64912164888576, 731420783788032, 8259345993203712, 93443504499523584, 1058972245409005568, 12019152955622817792, 136599995048232747008

Offset: 0

N. J. A. Sloane, Jan 23 2000

From Michel Marcus and Petros Hadjicostas, Jul 16 2020: (Start)
a(n) is the number of subdivisions of a 2 x n grid as defined in Robeva and Sun (2020). We have a(n) = A059576(n-1, n-1) for n >= 1 privided the latter is viewed as a square array (rather than a triangle).
In general, A059576(m-1, n-1) is the number of subdivisions of a 2-row grid with m points at the top row and n points at the bottom. (End)
The title condition is unclear: the path (0,0) -> (0,n) -> (n,n-1) -> (n,n) arguably meets the title condition but is not allowed, because steps with negative slope are proscribed. Steps must move east (slope 0) or have finite positive slope or move north (infinite slope). On the other hand, for lattice paths subject only to the condition that each successive point on the path is closer to the terminal point than its predecessor, see the question "Why are the numbers counting "ever-closer" lattice paths so round?" on the mathoverflow website. - David Callan, Nov 21 2021

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 6.3.9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020. See A(2,n) column in Table 3 (p. 10).

Main diagonal of A059576.
Column k=2 of A316674.
Cf. A084773, A316673.

[n eq 0 select 1 else 2^(n-1)*Evaluate(LegendrePolynomial(n), 3) : n in [0..40]]; // G. C. Greubel, May 21 2023

a[0]=1; a[n_]:= Hypergeometric2F1[-n, n+1, 1, -1]*2^(n-1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 23 2012, after Jon Stadler *)
Table[2^(n-1)*LegendreP[n,3] +Boole[n==0]/2, {n,0,40}] (* G. C. Greubel, May 21 2023 *)
CoefficientList[Series[(1+1/Sqrt[1-12x+4x^2])/2,{x,0,30}],x] (* Harvey P. Dale, Mar 10 2024 *)

def A052141(n): return 2^(n-1)*gen_legendre_P(n,0,3) + int(n==0)/2
[A052141(n) for n in range(41)] # G. C. Greubel, May 21 2023

G.f.: (1/2)*( 1 + 1/sqrt(1 - 12*x + 4*x^2) ).
a(n) = 2^(n-1) * A001850(n). - Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003
D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(8+6*sqrt(2))*(6+4*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012

A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 5, 23, 21, 5, 1, 1, 7, 79, 162, 66, 7, 1, 1, 11, 274, 1636, 1338, 192, 11, 1, 1, 15, 1003, 19977, 43686, 10585, 565, 15, 1, 1, 22, 3763, 298416, 2142277, 1178221, 82694, 1579, 22, 1, 1, 30, 14723, 5300296, 149056260, 232984145, 30370346, 612700, 4348, 30, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.

			Array begins:
============================================================
n\k | 0  1   2     3        4           5              6
----+-------------------------------------------------------
  0 | 1  1   1     1        1           1              1 ...
  1 | 1  1   2     3        5           7             11 ...
  2 | 1  2   7    23       79         274           1003 ...
  3 | 1  3  21   162     1636       19977         298416 ...
  4 | 1  5  66  1338    43686     2142277      149056260 ...
  5 | 1  7 192 10585  1178221   232984145    74676589469 ...
  6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...
  ...
The A(2,2) = 7 matrices are:
  [1 0]  [2 0]  [1 1]  [2 1]  [2 0]  [1 1]  [2 2]
  [1 0]  [0 1]  [1 0]  [0 1]  [0 2]  [1 1]
  [0 1]  [0 1]  [0 1]
  [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152

Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.
Columns k=0..3 are A000012, A000041, A331722, A331723.
Cf. A219727, A306017, A316674, A318951, A331315, A331461.

\\ See A318951 for RowSumMats
T(n, k)={RowSumMats(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

A306017(n) = Sum_{d|n} A(n/d, d).

A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 12, 4, 1, 0, 1, 124, 124, 8, 1, 0, 1, 1800, 10596, 1280, 16, 1, 0, 1, 33648, 1764244, 930880, 13456, 32, 1, 0, 1, 769336, 484423460, 1849386640, 85835216, 143808, 64, 1, 0, 1, 20796960, 198461691404, 7798297361808, 2098356708016, 8206486848, 1556416, 128, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.

A(n,k) is the number of n-uniform k-block sets of multisets.

			Array begins:
====================================================================
n\k | 0  1      2          3                4                   5
----+---------------------------------------------------------------
  0 | 1  1      0          0                0                   0 ...
  1 | 1  1      1          1                1                   1 ...
  2 | 1  2     12        124             1800               33648 ...
  3 | 1  4    124      10596          1764244           484423460 ...
  4 | 1  8   1280     930880       1849386640       7798297361808 ...
  5 | 1 16  13456   85835216    2098356708016  140094551934813712 ...
  6 | 1 32 143808 8206486848 2516779512105152 ...
  ...
The A(2,2) matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
  [1 0]  [0 2]  [0 1]  [0 1]  [0 2]  [1 2]
  [0 2]  [1 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..2 are A000012, A173219.
Columns k=0..2 are A000012, A011782, A331396.
The version with binary entries is A331277.
The version with not necessarily distinct columns is A331315.
Cf. A316674 (unrestricted version), A330942, A331395.

T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1,n),k)*sum(i=j, m, (-1)^(i-j)*binomial(i,j)))}

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).
A(n, k) = A011782(n) * A331277(n, k) for k > 0.
A331395(n) = Sum_{d|n} A(n/d, d).

A059516 Number of different relations between n intervals (possibly of zero length) on a line.

Original entry on oeis.org

1, 2, 26, 818, 47834, 4488722, 617364026, 117029670578, 29247820616474, 9318336983422802, 3686400233750527226, 1772940711343815069938, 1018732890334074636017114, 689260602916515788253448082, 542377407779814380777057527226, 491141384755777675851883392430898

Offset: 0

Henry Bottomley, Jan 19 2001

			a(1)=2 since if a is starting point of interval and A is end point then possibilities are aA (zero length) or a-A (positive length). a(2)=26 since possibilities are: aAbB, aAb-B, b-aAB, abB-A, a-AbB, ab-AB, aA-bB, bB-aA, aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB, a-A-b-B, a-b-A-B, a-b-B-A, b-B-a-A, b-a-B-A, b-a-A-B.

Alois P. Heinz, Table of n, a(n) for n = 0..224
IBM Ponder This, Jan 01 2001

Row n=2 of A316674. Row sums of A300729.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> add(b(n+k)*binomial(n, k), k=0..n)/2^n:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 10 2018

T[0, 0] = 1; T[n_, k_] := Sum[(-1)^(k-i) Binomial[k, i] (i(i+1)/2)^n, {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 1, 2n}]; a[0] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Oct 27 2020, from A300729 *)

a(n) = 2*A055203(n)-0^n = Sum_k A059515(k, n).
a(n) = Sum_{m>=0} binomial(m+1,2)^n/2^(m+1). a(n) = (1/2^n)*Sum_{k=0..n} binomial(n,k)*A000670(n+k). - Vladeta Jovovic, Aug 17 2006
E.g.f. as a continued fraction: 1/(1 + 2*(1 - exp(t))/(1 + 2*(1 - exp(2*t))/(1 + 2*(1 - exp(3*t))/(1 + ...)))) = 1 + 2*t + 26*t^2/2! + .... See A300729. - Peter Bala, Jun 13 2019

A316673 Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).

Original entry on oeis.org

1, 13, 818, 64324, 5592968, 515092048, 49239783968, 4831678931008, 483371425775744, 49083260519243008, 5043379069021557248, 523221884090930480128, 54715789513061864081408, 5760456190025868833542144, 609948004367577499751948288, 64905519628343663567453569024

Offset: 0

Alois P. Heinz, Jul 10 2018

Alois P. Heinz, Table of n, a(n) for n = 0..487

Column k=3 of A316674.

Cf. A052141, A126086.

a:= proc(n) option remember; `if`(n<4, [1, 13, 818, 64324][n+1],
     (2*(3*n-2)*(57*n^2-95*n+25)*a(n-1)-4*(9*n^3-30*n^2+29*n-6)*
      a(n-2)+8*(3*n-1)*(n-2)^2*a(n-3))/(n^2*(3*n-4)))
    end:
seq(a(n), n=0..20);

a[n_] := a[n] = If[n < 4, {1, 13, 818, 64324}[[n+1]], (2(3n-2)(57n^2- 95n+25) a[n-1] - 4(9n^3-30n^2+29n-6) a[n-2] + 8(3n-1)(n-2)^2 a[n-3]) / (n^2 (3n-4))];
a /@ Range[0, 20] (* Jean-François Alcover, May 14 2020, after Maple *)

Recurrence: see Maple program.
a(n) = A126086(n) * ceiling(2^(n-1)) = A126086(n) * A011782(n).
a(n) ~ sqrt((6 + 5*2^(1/3) + 4*2^(2/3))/6) * (24*2^(2/3) + 30*2^(1/3) + 38)^n / (4*Pi*n). - Vaclav Kotesovec, May 14 2020
G.f.: (1+hypergeom([1/3, 2/3],[1],108*x/(1-2*x)^3)/(1-2*x))/2. - Mark van Hoeij, Nov 28 2024

A316677 Number of paths from {0}^n to {n}^n that always move closer to {n}^n.

Original entry on oeis.org

1, 1, 26, 64324, 44418808968, 16811849850663255376, 5739186280165988958118118964128, 2625335736940471293251942853564104852964163648, 2236038557448505001455717540438942031299017768147674692322230400

Offset: 0

Alois P. Heinz, Jul 10 2018

Alois P. Heinz, Table of n, a(n) for n = 0..25

Main diagonal of A316674.

Cf. A011782, A262810.

a(n) = A316674(n,n).

a(n) = A262810(n) * ceiling(2^(n-1)) = A262810(n) * A011782(n).

A331636 Number of nonnegative integer matrices with nonzero rows, total sum n and columns with equal sums.

Original entry on oeis.org

1, 5, 17, 109, 557, 5785, 47357, 596365, 7151841, 106763725, 1622633597, 28757539337, 526858352477, 10758375762109, 230342261913169, 5344946955452877, 130370767029201437, 3394996816992588345, 92801587319328673277, 2681375183085608280093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

			The a(2) = 5 matrices are:
  [1]  [1 0]  [0 1]  [2]  [1 1]
  [1]  [0 1]  [1 0]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A316674.

a(n) = Sum_{d|n} A316674(n/d, d).

A337192 Triangular array read by rows. T(n,k) is the number of elements of rank k in the order complex of the poset P = [n] X [n], n=0, k=0 or n>0, 0<=k<=2n-1.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 2, 1, 9, 27, 37, 24, 6, 1, 16, 84, 216, 309, 252, 110, 20, 1, 25, 200, 800, 1875, 2751, 2570, 1490, 490, 70, 1, 36, 405, 2290, 7755, 17088, 25493, 26070, 18060, 8120, 2142, 252, 1, 49, 735, 5537, 25235, 76293, 160867, 242845, 264936, 207690, 114282, 41958, 9240, 924

Offset: 0

Geoffrey Critzer, Aug 18 2020

The poset P = [n] X [n] is the direct product of two chains of length n-1. The order complex of P is the set of all chains in P ordered by inclusion.

It appears that for n > 1, Sum_{k=0..2n-1} T(n,k) = 4*A052141(n-1). More generally, it appears that the number of elements in the order complex of [n]^k is four times the number of chains from bottom to top in [n]^k (Cf. A316674).

			  1,
  1, 1,
  1, 4,  5,   2,
  1, 9,  27,  37,  24,   6,
  1, 16, 84,  216, 309,  252,  110,  20,
  1, 25, 200, 800, 1875, 2751, 2570, 1490, 490, 70

Cf. A052141, A316674.

f[x_, y_] := If[x <= y, 1, 0];Prepend[CoefficientList[ 1 + z (Table[G = Array[f, {n, n}]; \[Zeta] = Level[Table[Table[Flatten[TensorProduct[G, G][[i]][[All, j]]], {j, 1, n}], {i, 1, n}], {2}];a = Inverse[IdentityMatrix[n^2] - z (\[Zeta] - IdentityMatrix[n^2])];Table[1, {n^2}].a.Table[1, {n^2}], {n, 1, 10}]),
   z], {1}] // Grid

cp's OEIS Frontend

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.

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A052141 Number of paths from (0,0) to (n,n) that always move closer to (n,n) (and do not pass (n,n) and backtrack).

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A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

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A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

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A059516 Number of different relations between n intervals (possibly of zero length) on a line.

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A316673 Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).

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