A059304 -id:A059304 - cp's OEIS frontend

A365643 Number of permutations whose reverse-complement shares the same recording tableau in the Robinson-Schensted correspondence.

1, 1, 2, 2, 12, 24, 136, 344, 2872, 7108, 80672, 211056, 3032376

Offset: 0

Dang-Son Nguyen, Sep 14 2023

This is an open problem from Martin's "Algebraic Combinatorics" lecture.

Tucker J. Ervin, Blake Jackson, Jay Lane, Kyungyong Lee, Son Dang Nguyen, Jack O'Donohue, and Michael Vaughan, Permutations whose reverse shares the same recording tableau in the RS correspondence, Sém. Lothar. Combin. 86 (2022), Art. B86a, 15 pp.
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, 2010-2023.
Wikipedia, Robinson-Schensted correspondence.

def a(n): return sum(StandardTableaux(T.shape()).cardinality()
               for T in StandardTableaux(n) if T == T.evacuation())
print([a(n) for n in range(13)])

A386670 Number of ternary strings of length 2*n that have more 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 1, 9, 73, 577, 4521, 35313, 275577, 2150721, 16793929, 131230609, 1026283545, 8032614625, 62921342953, 493262044977, 3869724080313, 30379987189377, 238661880787593, 1876072096450257, 14756076838714713, 116126703647975457, 914363729294862633, 7203083947383222897

Offset: 0

Enrique Navarrete, Jul 28 2025

nonn

			a(1)=1 since the string of length 2 is 00.
a(2)=9 since the strings of length 4 are the 4 permutations of 0001, the 4 permutations of 0002, and 0000.
a(4)=577 since the strings of length 8 are (number of permutations in parentheses): 00000001 (8), 00000002 (8), 00000011 (28), 00000012 (56), 00000022 (28), 00000111 (56), 00000112 (168), 00000122 (168), 00000222 (56), 00000000 (1).

Cf. A054554, A059304, A128417, A128418, A243019.

a(n) = Sum_{k=1..n} 2^(n-k)*binomial(2*n,n-k).
a(n) = Sum_{k=1..n} A128417(n,k).
G.f.: (1-4*x-sqrt(1-8*x))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A128418(n) - A059304(n).

A115902 Expansion of (1-8*x)^(-3/2).

Original entry on oeis.org

1, 12, 120, 1120, 10080, 88704, 768768, 6589440, 56010240, 472975360, 3972993024, 33228668928, 276905574400, 2300446310400, 19060840857600, 157569617756160, 1299949346488320, 10705465206374400, 88022713919078400, 722712809019801600, 5926245033962373120

Offset: 0

Paul Barry, Feb 02 2006

Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics 339.11 (2016): 2652-2659.

Cf. A002457, A059304.

a:= n-> add((binomial(2*n,n))*2^(n-2), j=1..n): seq(a(n), n=1..20); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[(1-8x)^-(3/2),{x,0,30}],x] (* Harvey P. Dale, Jul 13 2012 *)

G.f.: 1/((1-8*x)*sqrt(1-8*x)) = 1F0(3/2;;8x).
a(n) = Jacobi_P(n,1/2,1/2,1)*8^n.
a(n) = 2^n*(2*n+1)*binomial(2*n,n).
a(n) = (2*n+1)*A059304(n).
a(n) = 2^n*A002457(n).
D-finite with recurrence: n*a(n) -4*(2*n+1)*a(n-1) =0. - R. J. Mathar, Nov 14 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 8*x*(2*k+3)/(8*x*(2*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
Sum_{n>=0} (-1)^n/a(n) = 4/3*log(2). - Daniel Suteu, Oct 31 2017
Sum_{n>=0} 1/a(n) = 8*arcsin(1/sqrt(8))/sqrt(7). - Amiram Eldar, Jan 27 2024

A380611 Irregular triangle read by rows: T(r,c) is the product of the number of standard Young tableaux (A117506) and the number of semistandard Young tableaux (A262030) for partitions of r.

Original entry on oeis.org

1, 1, 3, 1, 10, 16, 1, 35, 135, 40, 45, 1, 126, 896, 875, 756, 375, 96, 1, 462, 5250, 10206, 8400, 2450, 14336, 2800, 875, 1701, 175, 1, 1716, 28512, 90552, 74250, 65856, 257250, 48000, 74088, 55566, 102900, 8100, 10976, 5488, 288, 1, 6435, 147147, 686400, 567567, 931392, 3244032, 606375, 194040, 2910600, 1448832, 2673000, 202125, 666792, 846720, 1029000, 491520, 19845, 24696, 65856, 14400, 441, 1

Offset: 0

Wouter Meeussen, Jan 28 2025

Partitions are generated in reverse lexicographic order.
Remark that A262030 uses Abramowitz-Stegun (A-St) order.
Sum of row r equals r^r for r > 0 (Robinson-Schensted correspondence).

			Triangle begins:
    1;
    1;
    3,    1;
   10,   16,     1;
   35,  135,    40,   45,    1;
  126,  896,   875,  756,  375,    96,    1;
  462, 5250, 10206, 8400, 2450, 14336, 2800, 875, 1701, 175, 1;
  ...
Fourth row is 1*35, 3*45, 2*20, 3*15, 1*1 with sum 256 = 4^4.

Wikipedia, Young tableau

Row sums give A000312.
Row lengths give A000041.
Leftmost column gives A088218.
Cf. A047874, A059304, A365643, A262030, A117506.

Needs["Combinatorica`"];
hooklength[par_?PartitionQ]:=Table[Count[par,q_/;q>=j]+1-i+par[[i]]-j,{i,Length[par]},{j,par[[i]]}];
countSYT[par_?PartitionQ]:=Tr[par]!/Times@@Flatten[hooklength[par]];
content[par_?PartitionQ]:=Table[j-i,{i,Length[par]},{j,par[[i]]}];
countSSYT[par_?PartitionQ,t_Integer_]:=Times@@((t+Flatten[content[par]])/Flatten[hooklength[par]]);
Table[countSYT[par] countSSYT[par,n],{n,8},{par,IntegerPartitions[n]}]

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A365643 Number of permutations whose reverse-complement shares the same recording tableau in the Robinson-Schensted correspondence.

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A386670 Number of ternary strings of length 2*n that have more 0's than the combined number of 1's and 2's.

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A115902 Expansion of (1-8*x)^(-3/2).

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Formula

A380611 Irregular triangle read by rows: T(r,c) is the product of the number of standard Young tableaux (A117506) and the number of semistandard Young tableaux (A262030) for partitions of r.

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Mathematica