A072167 T_10(n) in the notation of Bergeron et al., u_10(n) in the notation of Gessel: Related to Young tableaux of bounded height.

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916799, 479001478, 6227012074, 87177809092, 1307651456625, 20921799763626, 355647213494682, 6400805686152436, 121585553747301448, 2430677026538811240

Offset

1,2

Comments

In general, column k > 1 of A214015 is asymptotic to (Product_{j=1..k} j!) * k^(2*n + k^2/2) / (Pi^((k-1)/2) * 2^((k-1)*(k+2)/2) * n^((k^2-1)/2)). - Vaclav Kotesovec, Sep 10 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.

Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Formula

a(n) ~ 546852789 * 2^(2*n + 26)* 5^(2*n + 55) / (n^(99/2) * Pi^(9/2)). - Vaclav Kotesovec, Sep 10 2014

Recurrence: (n+9)^2*(n + 16)^2*(n + 21)^2*(n + 24)^2*(n + 25)*a(n) = 2*(110*n^9 + 13497*n^8 + 704088*n^7 + 20286926*n^6 + 350428790*n^5 + 3672058745*n^4 + 22314826244*n^3 + 68412531600*n^2 + 64584000000*n - 45705600000)*a(n-1) - 4*(n-1)^2*(4092*n^7 + 349536*n^6 + 12084237*n^5 + 215646686*n^4 + 2088552609*n^3 + 10433770264*n^2 + 21962910556*n + 8002290720)*a(n-2) + 8*(n-2)^2*(n-1)^2*(61160*n^5 + 3273732*n^4 + 64938154*n^3 + 580006399*n^2 + 2208119423*n + 2488711560)*a(n-3) - 256*(n-3)^2*(n-2)^2*(n-1)^2*(21076*n^3 + 577824*n^2 + 4743323*n + 11018760)*a(n-4) + 7372800*(n-4)^2*(n-3)^2*(n-2)^2*(n-1)^2*(2*n + 15)*a(n-5). - Vaclav Kotesovec, Sep 10 2014

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! / mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 10, []):
seq(a(n), n=0..25); # Vaclav Kotesovec, Sep 10 2014, after Alois P. Heinz

Mathematica

RecurrenceTable[{-7372800*(-4 + n)^2*(-3 + n)^2*(-2 + n)^2*(-1 + n)^2*(15 + 2*n)*a[-5 + n] + 256*(-3 + n)^2*(-2 + n)^2*(-1 + n)^2*(11018760 + 4743323*n + 577824*n^2 + 21076*n^3)*a[-4 + n]-8*(-2 + n)^2*(-1 + n)^2*(2488711560 + 2208119423*n + 580006399*n^2 + 64938154*n^3 + 3273732*n^4 + 61160*n^5)*a[-3 + n] + 4*(-1 + n)^2*(8002290720 + 21962910556*n + 10433770264*n^2 + 2088552609*n^3 + 215646686*n^4 + 12084237*n^5 + 349536*n^6 + 4092*n^7)*a[-2 + n]-2*(-45705600000 + 64584000000*n + 68412531600*n^2 + 22314826244*n^3 + 3672058745*n^4 + 350428790*n^5 + 20286926*n^6 + 704088*n^7 + 13497*n^8 + 110*n^9)*a[-1 + n] + (9 + n)^2*(16 + n)^2*(21 + n)^2*(24 + n)^2*(25 + n)*a[n]==0, a[1]==1, a[2]==2, a[3]==6, a[4]==24, a[5]==120}, a, {n, 1, 20}] (* Vaclav Kotesovec, Sep 10 2014 *)

Crossrefs

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=10 of A214015.

Sequence in context: A267390 A360463 A193937 * A230233 A154659 A254523

Adjacent sequences: A072164 A072165 A072166 * A072168 A072169 A072170

Keyword

nonn

Author

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 29 2002

Status

approved