2×7 格子内填入 a(i,j)∈{1,2,3}，a(i,j)≤a(i+1,j)，a(i,j)≤a(i,j+1)，有几种填法?

wintex

請問數學115117

王守恩

在7(列)×2(行)的方格内填1,2,3。每个方格恰好填1个数，要求左方格≤右方格,上方格≤下方格,有540种填法。
从简单开始。
1(列)×2(行)有001+001+01+01+1+1=006种填法。
2(列)×2(行)有006+005+03+03+2+1=020种填法。
3(列)×2(行)有020+014+06+06+3+1=050种填法。
4(列)×2(行)有050+030+10+10+4+1=105种填法。
5(列)×2(行)有105+055+15+15+5+1=196种填法。
6(列)×2(行)有196+091+21+21+6+1=336种填法。
7(列)×2(行)有336+140+28+28+7+1=540种填法。
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A002415——gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing.
6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736,- 埃里克·W·韦斯坦，2026年3月14日

在a(列)×b(行)的方格内填1,2,3,...,c。每个方格恰好填1个数，要求左方格≤右方格,上方格≤下方格,有S(a,b,c)种填法。
S(a,2,3){6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156,58870, 67425}——A002415,
S(a,3,3){10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575}——A006542,
S(a,4,3){15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, 26883780, 37823500, 52474500}——A006857,
S(a,5,3){21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, 582481900, 885069900}——A108679,
S(a,6,3){28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, 9735768900}——A134288,
S(a,2,4){10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626,}——A006542,
S(a,3,4){20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, 206108980, 302588000}——A047819,
S(a,4,4){35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460}——A107915,
S(a,5,4){56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480}——A140901,
S(a,6,4){84, 2520, 41580, 457380, 3737448, 24293412, 131589315, 614083470, 2530768240, 9386849472, 31803696288, 99604982880, 291153026880, 800670823920, 2085276513474, 5172303508911, 12276881393700}——A140903,

\(\displaystyle S(a,b,c)=\prod_{i=1}^a\prod_{j=1}^b\frac{i + j + c - 2}{i + j - 1}\) —— 可以有统一的通项公式——OEIS可没有这么干脆的！！！