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@rburing Here, there is a question on the Steenrod Algebra: https://ask.sagemath.org/question/365... I am also studying this problem. Can you calculate it in Sage? My Email: vochau150488@gmail.com

@rburing Do you study on the Steenrod algebra? I look forward to cooperating with you to write an article. I need the support of the software Sagemath.

Thank you very much!

Thank you very much! Now, for example set S_1:=S((1,1,1,3,4)); S_2:=S((1,2,3,4,5)); S_3:=S((1,1,1,6,6)). How to compute S_1+S_2+S_3 (mod 2) in Sage?

Yes, $\lambda_1\lambda_2\neq \lambda_2\lambda_1.$

Moreover, the monomial $\lambda_a\lambda_b\lambda_c\lambda_d\lambda_e$ is not ordere partitions. More presely, the subscripts $a, b, c, d, e$ are not necessary to order partition. For example, $S = \lambda_1\lambda_3\lambda_2^2\lambda_4\lambda_5 + \lambda_3\lambda_2^2\lambda_1\lambda_4\lambda_5 + \lambda_1\lambda_2^2\lambda_3\lambda_4\lambda_5.$

In the online version of Sagemath https://sagecell.sagemath.org/, for S((1,2,3,4,5)), it is not found the result.

Fix the positive integer numbers $t_1, t_2, t_3,t_4, t_5.$ We have the following formula:

$$ S= \sum_{i, j, h, m, k_1 + k_2+k_3+k_4 = i-t_1, \ell_1+\ell_2 + \ell_3 = j -t_2 + k_4, u_1 + u_2 = h - t_3+k_3+\ell_3 }M_1.M_2.M_3. M_4,$$ where

$$ M_1 = \binom{t_5-k_1}{k_1}\binom{t_4-k_2}{k_2}\binom{t_3-k_3}{k_3}\binom{t_2-k_4}{k_4}$$ $$ M_2 = \binom{t_5-k_1-\ell_1}{\ell_1}\binom{t_4-k_2-\ell_2}{\ell_2}\binom{t_3-k_3-\ell_3}{\ell_3}$$ $$ M_3 = \binom{t_5-k_1-\ell_1-u_1}{u_1}\binom{t_4-k_2-\ell_2-u_2}{u_2};$$ $$ M_4=\binom{t_1+t_2+t_3+t_4 +t_5-i - j-h-m}{m - t_4 + k_2+ \ell_2 + u_2}.\lambda_i\lambda_j\lambda_h\lambda_m\lambda_{t_1+t_2 + t_3+t_4+t_5 - i - j-h-m}$$ Here, the binomial factors $\binom{n}{k}$ mod 2 and the value of $S$ mod 2. By convention, $\binom{n}{k} \equiv 0$ (mod 2) if either $k < 0$ or $n < 0$ or $k > n.$

I don't how to construct this formula in SAGE. Can someone show me how to compute it using SAGE?