A369328 -id:A369328 - cp's OEIS frontend

A369326 Array read by ascending antidiagonals: A(n,k) is the number of words of length n over the alphabet [k] and sortable by a (2,1)-pop stack of depth 2.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 24, 16, 5, 1, 0, 1, 32, 59, 52, 25, 6, 1, 0, 1, 64, 138, 149, 95, 36, 7, 1, 0, 1, 128, 313, 396, 310, 156, 49, 8, 1, 0, 1, 256, 696, 1003, 923, 571, 238, 64, 9, 1, 0, 1, 512, 1527, 2458, 2585, 1884, 966, 344, 81, 10, 1

Offset: 0

Stefano Spezia, Jan 20 2024

			The array begins:
  1, 1,  1,   1,   1,   1, ...
  0, 1,  2,   3,   4,   5, ...
  0, 1,  4,   9,  16,  25, ...
  0, 1,  8,  24,  52,  95, ...
  0, 1, 16,  59, 149, 310, ...
  0, 1, 32, 138, 396, 923, ...
  ...

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.25 at page 13.

Cf. A000007 (k=0), A000012 (k=1 or n=0), A000079 (k=2).
Cf. A001477 (n=1), A000290 (n=2), A256857 (n=3).
Cf. A369324, A369327 (main diagonal), A369328 (antidiagonal sums).

A[n_,k_]:=SeriesCoefficient[((1-x)(1-2x)-((1-x)(1-2x)+x^2)y)/((1-x)(1-2x)-(1-x)(2-3x)y+(1-2x)y^2),{x,0,n},{y,0,k}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

G.f.: ((1 - x)(1 - 2*x) - ((1 - x)*(1 - 2*x) + x^2)*y)/((1 - x)*(1 - 2*x) - (1 - x)*(2 - 3*x)*y + (1 - 2*x)*y^2).

A372883 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric peaks, with k >= 0.

Original entry on oeis.org

1, 2, 4, 1, 9, 5, 23, 17, 1, 63, 51, 8, 176, 149, 39, 1, 491, 439, 153, 11, 1362, 1308, 540, 70, 1, 3762, 3912, 1812, 342, 14, 10369, 11671, 5935, 1439, 110, 1, 28559, 34637, 19175, 5541, 645, 17, 78653, 102222, 61302, 20214, 3170, 159, 1, 216638, 300190, 194080, 71242, 13903, 1089, 20

Offset: 1

Stefano Spezia, May 15 2024

			The irregular triangle begins:
     1;
     2;
     4,    1;
     9,    5;
    23,   17,    1;
    63,   51,    8;
   176,  149,   39,   1;
   491,  439,  153,  11;
  1362, 1308,  540,  70,  1;
  3762, 3912, 1812, 342, 14;
  ...
T(4,1) = 5 since there are 5 flattened Catalan words of length 4 with 1 symmetric peak: 0100, 0101, 0010, 0110, and 0121.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 21-22.

Cf. A007051 (row sums), A290900 (2nd column), A369328 (1st column), A371965, A372879, A372884.

T[n_,k_]:=SeriesCoefficient[x(1-x)(1-2x)/(1-5x+8x^2-5x^3-x^2y+2x^3y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,14},{k,0,Floor[(n-1)/2]}]//Flatten

G.f.: x*(1 - x)*(1 - 2*x)/(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2*x^3*y).

Sum_{k>=0} T(n,k) = A007051(n-1).

cp's OEIS Frontend

A369326 Array read by ascending antidiagonals: A(n,k) is the number of words of length n over the alphabet [k] and sortable by a (2,1)-pop stack of depth 2.

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