A006330 -id:A006330 - cp's OEIS frontend

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

0, 11, 12, 14, 133, 138, 143, 148, 155, 158, 160, 168, 179, 182, 188, 195, 198, 204, 208, 216, 227, 230, 236, 240, 248, 2057, 2066, 2071, 2077, 2084, 2091, 2094, 2101, 2106, 2111, 2120, 2131, 2134, 2140, 2149, 2154, 2159, 2164, 2171, 2174, 2192, 2211, 2214

Offset: 1

Gus Wiseman, Nov 17 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
    0: ()
   11: (2,1,1)
   12: (1,3)
   14: (1,1,2)
  133: (5,2,1)
  138: (4,2,2)
  143: (4,1,1,1,1)
  148: (3,2,3)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)
  160: (2,6)
  168: (2,2,4)
  179: (2,1,3,1,1)
  182: (2,1,2,1,2)
  188: (2,1,1,1,3)

These compositions are counted by A262977 up to 0's.
Except for 0, a subset of A345917.
The unreversed version is A348614.
The unreversed negative version is A349154.
The negative version is A349155.
A non-reverse unordered version is A349159, counted by A000712 up to 0's.
An unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[ Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==2*sats[stc[#]]&]

A115982 Number of planar partitions that are not corners.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 23, 54, 112, 228, 437, 826, 1499, 2685, 4688, 8079, 13668, 22875, 37738, 61676, 99672, 159742, 253681, 399962, 625741, 972756, 1502302, 2306988, 3522492, 5351239, 8088469, 12170163, 18229411, 27192571

Offset: 1

Alford Arnold, Feb 17 2006

a(n) can also be approximated by considering A000094 since A000094(n) = A000041(n) - n = 0 0 0 1 2 5 8 14 21 32 ... with partial sums 0 0 0 1 3 8 16 30 51 83 ... which counts many of the initial cases. The remaining cases form 0 0 0 0 0 2 7 24 ... counting for n=6, 22/11 and 21/21.

			The planar partitions begin 1 3 6 13 24 48 ... A000219 with corners 1 3 6 12 21 38 ... A006330; therefore the present sequence begins 0 0 0 1 3 10 ...

Cf. A000219, A006330, A000041, A000094.

a(n) = A000219(n) - A006330(n)

Edited with additional terms by Franklin T. Adams-Watters, Mar 10 2006

A288578 q-Expansion of wedge character chi^(2)(q).

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 55, 91, 151, 240, 381, 587, 900, 1352, 2018, 2966, 4332, 6250, 8962, 12725, 17962, 25147, 35015, 48414, 66603, 91071, 123945, 167786, 226154, 303375, 405337, 539249, 714740, 943659, 1241605, 1627812, 2127302, 2770966, 3598567

Offset: 0

N. J. A. Sloane, Jul 01 2017

nonn

Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.

Cf. A006330 (chi^(0)(q)), A001523 (chi^(1)(q)).

chi := proc(l,e)
    local gf,m,n,f;
    gf := 0 ;
    for m from 0 to e+1 do
        f := 1;
        for n from 1 to min(m+abs(l),e-m+1) do
            f := f/(1-q^n) ;
        end do:
        for n from 1 to min(m,e-m+1) do
            f := f/(1-q^n) ;
        end do:
        gf := gf+f*q^m ;
    end do:
    expand(gf) ;
    coeftayl(%,q=0,e) ;
end proc:
A288578 := proc(n)
    chi(2,n) ;
end proc:
for n from 0 do
    printf("%d,\n",A288578(n)) ;
end do: # R. J. Mathar, Jul 04 2017

A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 3, 4, 1, 4, 7, 4, 5, 1, 6, 10, 10, 5, 6, 1, 6, 14, 16, 13, 6, 7, 1, 8, 18, 26, 22, 16, 7, 8, 1, 8, 25, 34, 38, 28, 19, 8, 9, 1, 10, 29, 52, 55, 50, 34, 22, 9, 10, 1, 10, 37, 66, 84, 76, 62, 40, 25, 10, 11, 1, 12, 44, 90, 116, 122, 97, 74, 46, 28, 11, 12

Offset: 0

Paul D. Hanna, Jul 29 2005

Related to planar partitions.

			Triangle begins:
1;
1,2;
1,2,3;
1,4,3,4;
1,4,7,4,5;
1,6,10,10,5,6;
1,6,14,16,13,6,7;
1,8,18,26,22,16,7,8;
1,8,25,34,38,28,19,8,9;
1,10,29,52,55,50,34,22,9,10; ...
Row sums form A006330 (offset 1):
{1,3,6,12,21,38,63,106,170,272,422,653,...},
(planar partitions with only one row and one column).
G.f. of diagonal n, D_n(x), is generated from g.f. of
row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2:
D_3(x) = 1 + 4*x + 10*x^2 + 16*x^3 + 22*x^4 + ...
= (1 + 2*x + 3*x^2)/(1-x)^2 = R_2(x)/(1-x)^2;
D_4(x) = 1 + 6*x + 14*x^2 + 26*x^3 + 38*x^4 + ...
= (1+ 4*x+ 3*x^2+ 4*x^3)/(1-x)^2 = R_3(x)/(1-x)^2.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A006330 (row sums).

T[n_, k_] := T[n, k] = If[n < k || k < 0, 0, If[k == 0, 1, If[k == n, n + 1, Sum[T[n - k - 1, j]*(k - j + 1), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2017 *)

PARI
```
T(n,k)=if(n
				
```

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));if(n

T(n, k) = Sum_{j=0..k} T(n-k-1, j)*(k-j+1) with T(n, n) = n+1.

G.f.: A(x, y) = Sum_{j=0..n} x^j/Product_{i=1..j+1} (1-y*x^i)^2.

A116600 a(n) = A011782(n) + A000219(n) - A000712(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 15, 40, 103, 238, 531, 1131, 2362, 4811, 9694, 19307, 38243, 75400, 148443, 291984, 574724, 1132368, 2234617, 4416937, 8745567, 17343737, 34446090, 68500682, 136374947, 271755878, 541950747, 1081467319, 2159170372, 4312555339, 8616279482, 17219151572, 34418065540, 68805730450, 137566021077

Offset: 0

Alford Arnold, Feb 18 2006

Old definition was "Counts compositions plus plane partitions less partitions into parts of two kinds".

A116600 is essentially A115981 + A115982 since A000712 = A001523 + A006330.

			a(8) = 103 because A011782(8) + A000219(8) - A000712(8) = 128 + 160 - 185.

Cf. A000219, A000712, A001523, A006330, A011782, A115981, A115982.

N=66; x='x+O('x^N);
gf011782 =(1-x)/(1-2*x);
gf000219 = 1/prod(n=1,N, (1-x^n)^n );
gf000712 = 1/eta(x)^2;
Vec( gf011782 + gf000219 - gf000712 )
\\ Joerg Arndt, Mar 25 2014

a(n) = A011782(n) + A000219(n) - A000712(n).

Terms a(9) and beyond from Joerg Arndt, Mar 25 2014

A259100 Triangle read by rows, arising in the enumeration of corners.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 5, 2, 1, 0, 5, 8, 5, 2, 1, 0, 6, 14, 10, 5, 2, 1, 0, 7, 20, 18, 10, 5, 2, 1, 0, 8, 30, 30, 20, 10, 5, 2, 1

Offset: 0

N. J. A. Sloane, Jun 22 2015

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  2,  1;
  0,  4,  5,  2,  1;
  0,  5,  8,  5,  2,  1;
  0,  6, 14, 10,  5,  2,  1;
  0,  7, 20, 18, 10,  5,  2,  1;
  0,  8, 30, 30, 20, 10,  5,  2,  1;
  ...

G. Kreweras, Sur les extensions linéaires d'une famille particuliere d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)

Row sums = A006330.

Kreweras gives a recurrence on page 281.

A259101 Square array read by antidiagonals arising in the enumeration of corners.

Original entry on oeis.org

1, 2, 2, 5, 16, 5, 14, 91, 91, 14, 42, 456, 936, 456, 42, 132, 2145, 7425, 7425, 2145, 132, 429, 9724, 50765, 85800, 50765, 9724, 429, 1430, 43043, 315315, 805805, 805805, 315315, 43043, 1430, 4862, 187408, 1831648, 6584032, 9962680, 6584032, 1831648, 187408, 4862, 16796, 806208, 10127988, 48674808, 103698504, 103698504, 48674808, 10127988, 806208, 16796

Offset: 0

N. J. A. Sloane, Jun 22 2015

See Kreweras (1979) for precise definition.

			The first few antidiagonals are:
    1,
    2,    2,
    5,   16,    5,
   14,   91,   91,   14,
   42,  456,  936,  456,   42,
  132, 2145, 7425, 7425, 2145, 132,
  ...

G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)

The first row and column of the array are the Catalan numbers A000108.

The second row and column are A214824.

a[x_, y_] := (2(2x+2y+1)!(x^2+3x*y+y^2+4x+4y+3)) / (x!(x+1)!y!(y+1)!(x+y+1)(x+y+2)(x+y+3));
Table[Table[a[x-y, y], {y, 0, x}] // Reverse, {x, 0, 9}] // Flatten (* Jean-François Alcover, Aug 11 2017 *)

Kreweras gives an explicit formula for the general term (see bottom display on page 291).

More terms from Jean-François Alcover, Aug 11 2017

A356367 Number of plane partitions of n having exactly one row and one column, each of equal length.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 11, 16, 26, 36, 58, 81, 122, 172, 251, 350, 502, 692, 972, 1332, 1842, 2499, 3414, 4592, 6200, 8277, 11064, 14656, 19424, 25544, 33584, 43880, 57274, 74362, 96429, 124468, 160422, 205942, 263938, 337083, 429768

Offset: 0

Jeremy Lovejoy, Oct 16 2022

nonn

The empty plane partition of 0 contributes an initial term equal to 1.

Also equal to the number of unimodal compositions of n+1 where the peak appears exactly once and the number of parts to the left of the peak is equal to the number of parts to the right of the peak.

			For n = 7 the valid unimodal compositions of n+1=8 are (8), (1,6,1), (1,5,2), (2,5,1), (3,4,1), (1,4,3), (2,4,2), (1,1,4,1,1), (1,1,3,2,1), (1,2,3,1,1) and (1,1,1,2,1,1,1), and so a(7) = 11.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
B. Kim and J. Lovejoy, The rank of a unimodal sequence and a partial theta identity of Ramanujan, Int. J. Number Theory 10 (2014), 1081-1098.

Cf. A006330, A195012.

nmax = 50; CoefficientList[Series[1 + 1/Product[(1 - x^k)^2, {k, 1, nmax}] * Sum[(-1)^(k + r + 1) * x^(k*(k + 1)/2 + r*(r + 1)/2 + 2*k*r)*(1 - x^r), {r, 0, nmax}, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 28 2023 *)

G.f.: 1 + (1/Product_{n>=1}(1-x^n)^2)*Sum_{r,n>=0}(-1)^(n+r+1)*x^(n*(n+1)/2 + r*(r+1)/2 + 2*n*r)*(1-x^r).

cp's OEIS Frontend

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

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A115982 Number of planar partitions that are not corners.

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A288578 q-Expansion of wedge character chi^(2)(q).

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Maple

A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.

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A116600 a(n) = A011782(n) + A000219(n) - A000712(n).

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A259100 Triangle read by rows, arising in the enumeration of corners.

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A259101 Square array read by antidiagonals arising in the enumeration of corners.

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A356367 Number of plane partitions of n having exactly one row and one column, each of equal length.

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