a(n)/12 is the variance of the number of inversions of a random permutation of n letters. See evidence in Mathematica code below. - Geoffrey Critzer, May 15 2010
The sequence is related to A033487 by A033487(n-1) = n*a(n) - Sum_{i=0..n-1} a(i) = n*(n+1)*(n+2)*(n+3)/4. - Bruno Berselli, Apr 04 2012
Deleting the two 0's leaves row 2 of the convolution array A213750. - Clark Kimberling, Jun 20 2012
For n>=4, a(n-2) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0110 (the first n-4 zeros), or, the same, a(n-2) is up-down coefficient {n,6} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014
Minimum sum of the bottom row of a triangular array A filled with the integers [0..binomial(n, 2) - 1] that obeys the rule A[i, j] + 1 <= A[i+1, j] and A[i, j] + 1 <= A[i, j-1]. - C.S. Elder, Oct 13 2023
The preceding statement can be extended: a(n) is the minimum sum of the main antidiagonal of a n X n square array A filled eith the integers [0..n^2-1] that is increasing on each row from left to right, and on each column from top to bottom. - Yifan Xie, Dec 19 2024
REFERENCES
V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.
Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Sheila Sundaram, and Lei Xue, Topology of Cut Complexes II, arXiv:2407.08158 [math.CO], 2024. See p. 15.
