Timeline for MacMahon Master Theorem for non-matching coefficients

10 events

when toggle format	what		by	license	comment
Jul 13, 2022 at 4:43	vote	accept	Pluviophile
Jul 7, 2022 at 8:25	vote	accept	Pluviophile
Jul 7, 2022 at 8:25
Jun 29, 2022 at 18:19	vote	accept	Pluviophile
Jun 29, 2022 at 18:19
Jun 25, 2022 at 19:08	history	edited	Pluviophile		edited tags
Jun 25, 2022 at 18:49	history	edited	Pluviophile	CC BY-SA 4.0	added 96 characters in body; edited tags
Jun 25, 2022 at 8:28	answer	added	Pluviophile		timeline score: 10
Jun 24, 2022 at 18:52	comment	added	Max Alekseyev		This is not a generalization of MMT, but just a direct formula for the coefficient in question: $$[x_1^{p_1}\cdots x_n^{p_n}]\ X_1^{q_1}\cdots X_n^{q_n} = \sum_M \binom{q_1}{m_{11},\dots,m_{1n}}\cdots\binom{q_n}{m_{n1},\dots,m_{nn}},$$ where the sum is taken over all matrices $M=(m_{ij})_{i,j=1}^n$ with nonnegative integer entries, row sums $q_1,\dots,q_n$, and column sums $p_1,\dots,p_n$.
Jun 24, 2022 at 17:14	comment	added	Max Alekseyev		Given that there is no relation between $p_i$ and $q_i$ besides equal sums, and that $X_1^{q_1}\cdots X_n^{q_n}$ is a homogeneous polynomial in $x_i$, essentially you ask for a formula for the coefficients in the expansion of $X_1^{q_1}\cdots X_n^{q_n}$. This would be a very broad generalization of MMT, which focuses on a very particular coefficient.
Jun 24, 2022 at 12:28	history	edited	Pluviophile	CC BY-SA 4.0	deleted 8 characters in body
Jun 24, 2022 at 8:54	history	asked	Pluviophile	CC BY-SA 4.0