A001973 -id:A001973 - cp's OEIS frontend

A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6

Offset: 0

Henry Bottomley, Feb 17 2002

The number of partitions of m into up to n positive integers each no more than k is maximized for given n and k by m=floor(nk/2) or ceiling(nk/2) (and possibly some other values).

			Rows start:
1, 1, 1, 1, 1, 1, ...;
1, 1, 1, 1, 1, 1, ...;
1, 1, 2, 2, 3, 3, ...;
1, 1, 2, 3, 5, 6, ...;
1, 1, 3, 5, 8, 12, ...; etc.
T(4,5)=12 since 10 can be partitioned into
5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3,
4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, and 3+3+2+2.

As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.

A067059 := proc(n,k)
    local m,a1,a2 ;
    a1 := 0 ;
    m := floor(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a1 := a1+1 ;
            end if ;
        end if;
    end do:
    a2 := 0 ;
    m := ceil(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a2 := a2+1 ;
            end if ;
        end if;
    end do:
    max(a1,a2) ;
end proc:
for d from 0 to 12 do
    for k from 0 to d do
        printf("%d,",A067059(d-k,k)) ;
    end do:
end do: # R. J. Mathar, Nov 13 2016

t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 02 2012 *)

def A067059(n, k):
    return Partitions((n*k)//2, max_length=n, max_part=k).cardinality()
for n in (0..9): [A067059(n,k) for k in (0..9)] # Peter Luschny, May 05 2014

A201496 T(n,k)=Number of nXk zero-sum -2..2 arrays with rows and columns lexicographically nondecreasing.

Original entry on oeis.org

1, 3, 3, 5, 29, 5, 8, 226, 226, 8, 12, 1454, 9331, 1454, 12, 18, 7815, 301432, 301432, 7815, 18, 24, 36487, 7798656, 50432052, 7798656, 36487, 24, 33, 151593, 168357815, 6697376240, 6697376240, 168357815, 151593, 33, 43, 571539, 3127432827

Offset: 1

R. H. Hardin Dec 02 2011

Table starts
..1.......3............5................8..................12
..3......29..........226.............1454................7815
..5.....226.........9331...........301432.............7798656
..8....1454.......301432.........50432052..........6697376240
.12....7815......7798656.......6697376240.......4585552158911
.18...36487....168357815.....732383803600....2571822263246639
.24..151593...3127432827...68110029402413.1219998885223727883
.33..571539..51118964623.5516712973977104
.43.1982705.747555368709
.55.6399842

			Some solutions for n=3 k=3
.-2..1..2...-1..1..1...-2.-1..0...-2.-1..0...-2.-2.-1...-2.-2..1....0..0..1
..0..0..1....1.-2.-1....0.-1..2...-1..2.-1...-2..2..2...-1..0..0....0..1.-1
..2.-2.-2....2..1.-2....2..2.-2....1..1..1....0..1..2....1..1..2....2.-1.-2

R. H. Hardin, Table of n, a(n) for n = 1..71

Column 1 is A001973

A183917 T(n,k) = number of nondecreasing arrangements of n numbers in -k..k with sum zero.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 3, 1, 5, 8, 8, 3, 1, 6, 13, 18, 12, 4, 1, 7, 18, 33, 32, 18, 4, 1, 8, 25, 55, 73, 58, 24, 5, 1, 9, 32, 86, 141, 151, 94, 33, 5, 1, 10, 41, 126, 252, 338, 289, 151, 43, 6, 1, 11, 50, 177, 414, 676, 734, 526, 227, 55, 6, 1, 12, 61, 241, 649, 1242, 1656, 1514

Offset: 1

R. H. Hardin, Jan 07 2011

			Table starts
 1  1   1    1    1     1     1      1      1      1       1       1       1
 2  3   4    5    6     7     8      9     10     11      12      13      14
 2  5   8   13   18    25    32     41     50     61      72      85      98
 3  8  18   33   55    86   126    177    241    318     410     519     645
 3 12  32   73  141   252   414    649    967   1394    1944    2649    3523
 4 18  58  151  338   676  1242   2137   3486   5444    8196   11963   17002
 4 24  94  289  734  1656  3370   6375  11322  19138   30982   48417   73316
 5 33 151  526 1514  3788  8512  17575  33885  61731  107233  178870  288100
 5 43 227  910 2934  8150 20094  45207  94257 184717  343363  610358 1043534
 6 55 338 1514 5448 16660 44916 109583 246448 517971 1028172 1943488 3521260
Some solutions for n=5:
  -2  -4  -4  -4  -4  -1  -4  -3  -4  -3  -1  -4  -3  -3  -2  -4
  -2   0   0  -1  -2   0  -2  -2  -1  -3  -1  -4   0  -2   0  -3
   0   0   0   0  -1   0   1  -1   1   0   0   1   0   1   0  -1
   0   1   2   2   3   0   2   3   2   3   0   3   0   1   1   4
   4   3   2   3   4   1   3   3   2   3   2   4   3   3   1   4

R. H. Hardin, Table of n, a(n) for n = 1..1350
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Definition 5.1, p. 8.
Karlee J. Westrem, Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group, Ph. D. Dissertation, Michigan Tech. Univ. (2025). See p. 55.

Column 2 is A001973.
Column 3 is A001977.
Column 4 is A001981.
Diagonal is A109655.
Row 3 is A000982(n+1).

from sympy.utilities.iterables import partitions
def A183917_T(n,k): return sum(1 for p in partitions(k*n,m=n,k=k<<1)) # Chai Wah Wu, Aug 27 2024

A201503 T(n,k)=Number of nXk 0..1 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 3, 6, 3, 2, 1, 3, 5, 5, 3, 1, 2, 4, 12, 8, 12, 4, 2, 1, 4, 8, 12, 12, 8, 4, 1, 2, 5, 20, 18, 40, 18, 20, 5, 2, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 2, 6, 30, 33, 98, 58, 98, 33, 30, 6, 2, 1, 6, 18, 43, 73, 94, 94, 73, 43, 18, 6, 1, 2, 7, 42, 55, 204, 151, 338, 151, 204

Offset: 1

R. H. Hardin Dec 02 2011

Table starts
.2.1..2..1...2...1...2....1....2....1....2.....1.....2.....1.....2......1
.1.2..2..3...3...4...4....5....5....6....6.....7.....7.....8.....8......9
.2.2..6..5..12...8..20...13...30...18...42....25....56....32....72.....41
.1.3..5..8..12..18..24...33...43...55...69....86...104...126...150....177
.2.3.12.12..40..32..98...73..204..141..380...252...650...414..1042....649
.1.4..8.18..32..58..94..151..227..338..480...676...920..1242..1636...2137
.2.4.20.24..98..94.338..289..936..734.2234..1656..4770..3370..9344...6375
.1.5.13.33..73.151.289..526..910.1514.2430..3788..5744..8512.12346..17575
.2.5.30.43.204.227.936..910.3334.2934.9936..8150.25908.20094.60882..45207
.1.6.18.55.141.338.734.1514.2934.5448.9686.16660.27718.44916.70922.109583

			Some solutions for n=5 k=4
..0..0..0..1....0..0..0..1....0..0..1..1....0..0..0..0....0..0..0..0
..0..0..0..1....0..0..0..1....0..0..1..1....0..0..0..0....0..0..1..1
..0..0..1..1....0..0..0..1....0..0..1..1....0..0..1..1....0..0..1..1
..0..1..1..1....0..1..1..1....0..0..1..1....1..1..1..1....0..0..1..1
..0..1..1..1....1..1..1..1....0..0..1..1....1..1..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..3037

Column 2 is A004526(n+2)
Column 3 odd terms are A002378((n+1)/2)
Column 3 even terms are A000982((n+2)/2)
Column 4 is A001973
Column 5 even terms are A188183((n-2)/2)
Column 6 is A001977
Column 7 even terms are A188185((n-4)/2)
Column 8 is A001981
Column 10 is A183913
Column 12 is A183914
Column 14 is A183915
Column 16 is A183916

A224838 Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 4, 6, 5, 1, 1, 10, 10, 6, 1, 1, 5, 20, 15, 7, 1, 6, 15, 35, 21, 8, 1, 1, 21, 35, 56, 28, 9, 1, 1, 7, 56, 70, 84, 36, 10, 1, 8, 28, 126, 126, 120, 45, 11, 1, 1, 36, 84, 252, 210, 165, 55, 12, 1, 1, 9, 120, 210, 462, 330, 220, 66, 13, 1

Offset: 1

John Molokach, Jul 21 2013

The successive rows have lengths 1,2,2; 3,4,4; 5,6,6; 7,8,8; ...
Sum of row n is A005314(n).
Old definition was: "Triangle of falling diagonals of A011973 (with rows displayed as centered text)."

			First 11 rows of the triangle:
  1;
  1,  1;
  2,  1;
  1,  3,  1;
  1,  3,  4,  1;
  4,  6,  5,  1;
  1, 10, 10,  6,  1;
  1,  5, 20, 15,  7,  1;
  6, 15, 35, 21,  8,  1;
  1, 21, 35, 56, 28,  9,  1;
  1,  7, 56, 70, 84, 36, 10,  1;

N. J. A. Sloane, Construction of present triangle by reading triangle A011973 from right to left along the paths indicated.

Cf. A005314, A227300, A001973, A000045, A004396.

Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* T. D. Noe, Jul 25 2013 *)
Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* John Molokach, Jul 25 2013 *)

r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), k = 1..floor((2n+2)/3).

R(n) = binomial(n-floor((k+1)/2), n-floor((3k-1)/2)), k = 1..floor((2n+2)/3), gives the terms of each row in reverse order.

Entry revised by N. J. A. Sloane, Jul 07 2024

A331545 Triangle of constant term of the symmetric q-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 0, 4, 0, 8, 0, 4, 0, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 0, 5, 0, 18, 0, 18, 0, 5, 0, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 0, 6, 0, 33

Offset: 0

Michael Somos, Jan 19 2020

nonn

Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).

			Triangle begins:
  n\k| 0 1 2 3 4 5 6 7  ...
  ---+----------------
   0 | 1
   1 | 1 1
   2 | 1 0 1
   3 | 1 1 1 1
   4 | 1 0 2 0 1
   5 | 1 1 2 2 1 1
   6 | 1 0 3 0 3 0 1
   7 | 1 1 3 5 5 3 1 1
   ...

CF. A000982, A001973, A063074, A188181.

T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];

{T(n, k) = if( k<0 || k>n, 0, polcoeff( prod(j = 1, k, (x^(n+1-j) - x^(-n-1+j))/(x^j - x^(-j))), 0))};

T(2*n, 2*k+1) = 0. T(2*n+1, 3) = A000982(n). T(2*n+1, 5) = A001973(n) if n>=2. T(4*n, 2*n) = A063074(n).

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A067059 Square array read by antidiagonals of partitions which half fill an n*k box, i.e., partitions of floor(nk/2) or ceiling(nk/2) into up to n positive integers, each no more than k.

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A201496 T(n,k)=Number of nXk zero-sum -2..2 arrays with rows and columns lexicographically nondecreasing.

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A183917 T(n,k) = number of nondecreasing arrangements of n numbers in -k..k with sum zero.

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A201503 T(n,k)=Number of nXk 0..1 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

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A224838 Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.

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A331545 Triangle of constant term of the symmetric q-binomial coefficients.

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