

A291908


Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4*n1,4*n2,...,2,1) and mu is the square n^n.


1




OFFSET

0,2


COMMENTS

The number of standard Young tableaux of a fixed skew shape has a determinantal formula, the JacobiTrudi formula. It is rare when a family of skew shapes has a product formula for the number of standard Young tableaux. This product formula has independently been proved using PSchur functions (by DeWitt) and using the Naruse hooklength formula for skew shapes (by Morales, Pak and Panova).


LINKS

Table of n, a(n) for n=0..4.
E. A. DeWitt, Identities Relating Schur sFunctions and QFunctions, Ph.D. thesis, University of Michigan, 2012, 73 pp.
A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.


FORMULA

a(n) = (binomial(4*n,2)n^2)!*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)) where b(n) = 1!*2!*...*(n1)! is the superfactorial A000178(n1), and c(n) = 1!!*3!!*...*(2*n3)!! is super doublefactorial A057863(n1).


EXAMPLE

a(1)=16 since there are 16 standard Young tableaux of skew shape 321/1 since this is the same as the number of standard Young tableaux of straight shape 321 given by the hooklength formula: 16 = 6!/(3^2*5).


MAPLE

b:=n>mul(factorial(i), i=1..n1):
c:=n>mul(doublefactorial(2*i1), i=1..n1):
a:=n>factorial(binomial(4*n, 2)n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)):
seq(a(n), n=0..9);


PROG

(Sage)
def b(n): return mul([factorial(i) for i in range(1, n)])
def d(n): return factorial(n+1)/(2^((n+1)/2)*factorial((n+1)/2))
def c(n): return mul([d(2*i1) for i in range(1, n)])
def a(n):
return factorial(binomial(4*n, 2)n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n))
[a(n) for n in range(10)]


CROSSREFS

Cf. A000178, A057863, A008793, A291871, A000085, A061343, A005118.
Sequence in context: A323336 A013878 A058418 * A059933 A002488 A330716
Adjacent sequences: A291905 A291906 A291907 * A291909 A291910 A291911


KEYWORD

nonn


AUTHOR

Alejandro H. Morales, Sep 05 2017


STATUS

approved



