ABSTRACT I introduce k-crossing paths and partitions and count the number of paths for each number of desired crossings k for systems with 11 points or less. I give some conjectures into the number of possible paths for certain numbers of crossings as a function of the number of points. INTRODUCTION A order n meandric partition is a set of the integers 1⋯n, such that a path from the south-west can weave through n points labeled 1⋯n without intersecting itself and finally heads east (examples are shown in Fig. 1). Counting the number of possible paths for n points is a tricky problem, and no recursion relation, generating function or explicit formula for the number of order n meandric partitions appears to have been found. This work is concerned with the number of paths that must intersect themselves exactly k times, where when k is 0, we have the meandric paths. It is possible to draw a line that deliberately crosses itself as many times as required, because of this we only consider a path to be k-crossing if k is the smallest number of crossings possible, that is a path that must cross itself k times (an example of a 3-crossing path over 9 points is given in Fig. 2). RESULTS Define ak(n) to be the number of configurations of n points where the path through them is forced to cross itself k times. For 0-crossings on n points we have the open meandric numbers, given in the OEIS as A005316 a_0(n) = 1, 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, \cdots, \;\; n=0,1,\cdots this work has counted this for k > 0 by calculating all n! permutations of the n integers and checking to see the minimal number of crossings for each, we then have n =&0&1&2&3&4&5&6&7&8&9&10&11\cdots\\ a_0(n) =&1,& 1,& 1,& 2,& 3,& 8,& 14,& 42,& 81,& 262,& 538,& 1828,\cdots\\ a_1(n) =&0,&0,& 1,& 4,& 10,& 36,& 85,& 312,& 737,& 2760,& 6604, &25176,\cdots\\ a_2(n) =&0,&0,& 0,& 0,& 8,& 42,& 168,& 760,& 2418,& 10490,& 30842, &131676,\cdots\\ a_3(n) =&0,&0,& 0,& 0,& 2,& 16,& 164,& 944,& 4386,& 22240,& 83066, &398132,\cdots\\ a_4(n) =&0,&0,& 0,& 0,& 1,& 18,& 146,& 1076,& 6255,& 37250,& 168645, &908898,\cdots\\ a_5(n) =&0,&0,& 0,& 0,& 0,& 0,& 96,& 960,& 7388,& 51968,& 282122, &1711824, \cdots\\ a_6(n) =&0,&0,& 0,& 0,& 0,& 0,& 30,& 440,& 6472,& 55140,& 384065, &2642444,\cdots\\ a_7(n) =&0,&0,& 0,& 0,& 0,& 0,& 14,& 368,& 5176,& 53920,& 455944, &3575040,\cdots\\ a_8(n) =&0,&0,& 0,& 0,& 0,& 0,& 2,& 66,& 3542,& 45960,& 484058, &4336734,\cdots\\ a_9(n) =&0,&0,& 0,& 0,& 0,& 0,& 1,& 72,& 2011,& 32280,& 452504, &4661756,\cdots\\ a_{10}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1172,& 25066,& 396493, &4709856,\cdots\\ a_{11}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 420,& 11840,& 309696, &4291440,\cdots\\ a_{12}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 201,& 8930,& 225754, &3661348,\cdots\\ a_{13}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 40,& 2240,& 151849, &2947392,\cdots\\ a_{14}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 18,& 2040,& 91147, &2103648,\cdots\\ a_{15}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 2,& 224,& 55030, &1575744,\cdots\\ a_{16}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1,& 270,& 26762, &915924,\cdots\\ a_{17}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 14627, &665088,\cdots\\ a_{18}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 5405, &295956,\cdots\\ a_{19}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2642, &218508,\cdots\\ a_{20}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 641, &63522,\cdots\\ a_{21}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 293, &54672,\cdots\\ a_{22}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 48, &8964,\cdots\\ a_{23}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 22, &9552,\cdots\\ a_{24}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2, &706,\cdots\\ a_{25}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 1, &972,\cdots where the vertical sum over columns of terms gives n!. CONJECTURES The above information has lead to a few conjectures. a_{n^2}(2n) = 1 this can be converted to words as, there is exactly one path through 2n points that crosses n² times. The partitions associated with these paths are (2,1)\\ (3,1,4,2)\\ (4,1,5,2,6,3)\\ (5,1,6,2,7,3,8,4)\\ (6,1,7,2,8,3,9,4,10,5) and a clear interlaced pattern can be seen (an example is given in Fig. 3). a_{n^2-1}(2n) = 2, \; n>1 a_{n^2-2}(2n) = 4n+2, \; n>2 a_{n^2-3}(2n) = 8n+8, \; n>3 a_{n^2}(2n+1) = 2(n+1)3^{n-1}, \; n>1