A049401 Number of Young tableaux of height <= 5.

1, 1, 2, 4, 10, 26, 75, 225, 715, 2347, 7990, 27908, 99991, 365587, 1362310, 5159208, 19831101, 77233517, 304423574, 1212962072, 4881181036, 19821471956, 81165639197, 334925706659, 1391935877463, 5823186349671, 24511802558326, 103772782048252, 441696903185704

Offset: 0

Author

Alford Arnold, N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also the number of n-length words w over alphabet {a,b,c,d,e} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c) >= #(z,d) >= #(z,e), where #(z,x) counts the letters x in word z. The a(5) = 26 words are: aaaaa, aaaab, aaaba, aabaa, abaaa, aaabb, aabab, abaab, aabba, ababa, aaabc, aabac, abaac, aabca, abaca, abcaa, aabbc, ababc, aabcb, abacb, abcab, aabcd, abacd, abcad, abcda, abcde. - Alois P. Heinz, May 30 2012

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), y_5(n), p. 452.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
• F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
• Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017.
• Dominique Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, European J. Combin., 10(1):69-82, 1989.
• Alon Regev, Amitai Regev, Doron Zeilberger, Identities in character tables of S_n, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
• Index entries for sequences related to Young tableaux.

Crossrefs

Sum of first five diagonals of A047884. Cf. A007579.

Column k=5 of A182172. - Alois P. Heinz, May 30 2012

Programs

Maple

a:= proc(n) option remember;
      `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*a(n-1)
       +(n-1)*(13*n+9)*a(n-2) -15*(n-1)*(n-2)*a(n-3)) /
       ((n+4)*(n+6)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

a[n_] := a[n] = If[n<3, {1, 1, 2}[[n+1]], ((3*n^2+17*n+15)*a[n-1] + (n-1)*(13*n+9)*a[n-2] - 15*(n-1)*(n-2)*a[n-3]) / ((n+4)*(n+6))]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

PARI

a(n) = 6*n!*sum(k=0, n\2, (2*k+2)!/((n-2*k)!*k!*(k+1)!*(k+2)!*(k+3)!)); \\ Seiichi Manyama, Mar 27 2025

Formula

E.g.f.: e^x*(BesselI(0, 2*x)^2 - BesselI(0, 2*x)*BesselI(2, 2*x) - BesselI(0, 2*x)*BesselI(4, 2*x) - BesselI(1, 2*x)^2 + 2*BesselI(1, 2*x)*BesselI(3, 2*x) + BesselI(2, 2*x)*BesselI(4, 2*x) - BesselI(3, 2*x)^2) (BesselI = modified Bessel function of first kind).

a(n) ~ 3*5^(n+5)/(8*Pi*n^5). - Vaclav Kotesovec, Aug 18 2013

D-finite with recurrence (n+6)*(n+4)*a(n) +(-3*n^2-17*n-15)*a(n-1) -(13*n+9)*(n-1)*a(n-2) +15*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 23 2021

a(n) = 6 * n! * Sum_{k=0..floor(n/2)} (2*k+2)!/((n-2*k)!*k!*(k+1)!*(k+2)!*(k+3)!). - Seiichi Manyama, Mar 27 2025

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001